Every snapshot says vortex, and every particle leaves

The picture belongs to whoever is watching shows that the Q-criterion — rotation beating strain in the velocity gradient — changes with the observer: a flow with no rotation in it contains a vortex for anybody spinning faster than its strain rate. Where a vortex stops then finds that no criterion read off a snapshot can say which fluid stays together, because that is a question about trajectories over time.

Both essays leave room for a comfortable reading: the criteria are frame-dependent and threshold-dependent, but in the laboratory frame, for a vortex strong enough to leave no doubt, they surely say something true about the fluid. This essay closes that room. There is a flow in which the snapshot says vortex, emphatically and without interruption, in the frame the flow is measured in, and the fluid does the opposite.

A vortex, at every instant #

Take a velocity field that is linear in position, $\mathbf u = \mathsf A(t)\,\mathbf x$ — the flow near any point, to first order — with a gradient made of two parts. One is a strain of rate $s$ whose principal axes turn at a steady rate $\Omega_s$. The other is a uniform rotation, vorticity $\omega$. George Haller’s example of 2005 takes $s = 1$, $\Omega_s = -2$ and $\omega = -4$:

$$\mathsf A(t) = \begin{pmatrix} \sin 4t & 2 + \cos 4t \\ -2 + \cos 4t & -\sin 4t \end{pmatrix}.$$

At every instant its trace is zero, so it is incompressible, and its determinant is $4 - 1 = 3$, so its eigenvalues are $\pm i\sqrt3$. The Q-criterion, which in two dimensions is the determinant, is 3. The Okubo–Weiss parameter, the swirling strength and the $\lambda_2$ criterion all read the same eigenvalues and all agree. Checked over a whole period at four hundred instants, the determinant is 3 to $10^{-15}$. By every test that reads the velocity gradient at an instant, this is a vortex everywhere and always.

Release particles on a small circle about the origin and follow them. They do not circulate. Each spirals outward, turning as it goes, and its distance from the centre grows exponentially. After 2.4 time units the circle of radius 0.3 has been stretched to a curve reaching eleven times further out, and still accelerating. The flow is a vortex in every snapshot, and a fluid particle placed in it is flung away.

What the snapshot sees, and what the particles do #

The disagreement is total, not marginal. Q is not hovering near zero; it is three, larger than the square of the strain rate by a factor of three. And the particles do not drift slowly; they separate at the rate a saddle of strain rate one would separate them, doubling their distance every 0.69 time units.

The integration that shows it is checked against an exact result. The gradient repeats every $\pi/2$ time units, so the flow over one period is a fixed linear map, the monodromy matrix, and its eigenvalues decide everything about the long-time behaviour. Computed by fourth-order Runge–Kutta, it is $-\mathrm{diag}(e^{\pi/2}, e^{-\pi/2})$ to seven parts in $10^{14}$: multipliers of −4.810 and −0.208, one stretching and one compressing, a saddle. The finite-time Lyapunov exponent — the logarithm of the flow map’s largest stretching divided by the time — is exactly one over every interval.

Turning with the strain #

The explanation is in the frame the strain’s axes turn in.

Watch the flow from a frame that turns with the strain’s principal axes. In that frame the strain is steady. The fluid’s vorticity is reduced by twice the frame’s rotation rate, $\omega' = \omega - 2\Omega_s$, which for Haller’s numbers is $-4 + 4 = 0$. What is left is a pure, steady strain, and a pure strain is a saddle: the particles come in along the compressing axis and leave along the stretching one, on hyperbolas. The spirals in the laboratory frame are that saddle, turned.

The snapshot reads the vorticity relative to the observer. The fluid’s fate depends on the vorticity relative to the strain’s axes. When the axes stand still the two are the same. When the axes turn, the fluid feels a strain that is continually being rotated towards a new direction, and whether it circulates or escapes depends on how its own rotation compares with the axes’ — which no single instant records, because an instant has no turning rate in it.

This is not the frame-dependence of the earlier essay. There the observer’s rotation changed the criterion’s verdict. Here the observer is not rotating at all; the criterion is evaluated in the laboratory frame and is simply wrong about the fluid. The rotating frame is used only to explain why, and the verdict that matters — the particles escape — is the same in every frame, because distances between particles do not depend on who measures them.

The arithmetic of the turning frame #

The rotating-frame argument is two lines, and it is worth having exactly because it says which quantity the snapshot is missing. Write the position in the turning frame, $\mathbf x = \mathsf R(\Omega_s t)\,\mathbf y$. Then

$$\dot{\mathbf y} = \left(\mathsf R^{\mathsf T}\mathsf A\,\mathsf R - \Omega_s\mathsf J\right)\mathbf y = \left[\begin{pmatrix} s & 0 \\ 0 & -s \end{pmatrix} + \left(\tfrac{\omega}{2} - \Omega_s\right)\mathsf J\right]\mathbf y,$$

with $\mathsf J$ the quarter-turn. The strain comes out fixed along the axes, the fluid’s rotation is reduced by the frame’s, and the matrix no longer depends on time. Its eigenvalues are $\pm\sqrt{s^2 - (\omega/2 - \Omega_s)^2}$: real, and the particles escape, when the rotation left over is smaller than the strain; imaginary, and they circulate, when it is larger.

The snapshot computes the same expression with $\Omega_s$ set to zero, because at one instant the strain’s axes have a direction but no rate of turning. The whole of the error is the term the snapshot cannot see, and it enters doubled: a strain whose axes turn at $\Omega_s$ acts on the fluid as if the fluid’s own vorticity were $2\Omega_s$ smaller. For Haller’s flow that correction, $2 \times (-2) = -4$, exactly cancels the vorticity of $-4$.

A flow with no vorticity that holds its fluid #

The other pair of wedges is as surprising as the first. Set the vorticity to zero, so that the flow is a pure strain at every instant — irrotational, $Q = -s^2$ everywhere, a saddle to every snapshot — and let its axes turn at twice the strain rate. In the turning frame the fluid now carries a relative vorticity of $-2\Omega_s = -4$, twice the strain, and it circulates.

Followed in the laboratory frame, a particle released at a distance of 0.3 from the centre stays between 0.245 and 0.424 for as long as it is integrated, tracing a looping orbit that never escapes, and the Lyapunov exponent falls to $2 \times 10^{-3}$ over twenty time units. A flow with no vorticity anywhere, at any instant, holds every particle placed in it. It is not a vortex by any criterion that looks at the gradient, and it traps fluid as firmly as a vortex does.

The same mechanism holds a ball on a saddle-shaped surface spun fast enough — the ball rolls downhill along one direction, but the direction keeps turning away before it has gone far — and, in electric rather than gravitational fields, holds a single ion in a Paul trap. In a fluid it appears wherever a strain rotates faster than it stretches, which a snapshot will always report as a place where the fluid is being pulled apart.

Where the snapshot is right #

The example generalises into a map. Any flow of this family is labelled by two numbers, the vorticity and the turning rate of the strain’s axes, each divided by the strain rate. The snapshot calls a point a vortex when $|\omega| > 2s$. The fluid circulates when $|\omega - 2\Omega_s| > 2s$ and escapes, at the rate $\sqrt{s^2 - (\omega - 2\Omega_s)^2/4}$, when not.

The two conditions are two bands in the plane, one vertical and one slanted, and they disagree in two pairs of wedges. In one pair the snapshot sees a vortex and the particles escape: false vortices, of which Haller’s flow is one. In the other the snapshot sees a saddle and the particles circulate: hidden vortices, where a strain whose axes turn fast enough in the same sense as the fluid holds it rather than flinging it — the principle a rotating saddle-shaped surface uses to trap a rolling ball. The bands cross on the horizontal axis, and along it, where the strain’s axes do not turn, the snapshot’s verdict and the fluid’s agree everywhere.

That last fact is the useful one. An instantaneous vortex criterion is exact for flows whose strain does not rotate, and fails in proportion to how fast the strain’s axes turn. In a steady flow the axes at a point do not turn, and the criteria are right about the fluid as well as the gradient. Where the flow is unsteady in a way that rotates the strain — near a vortex being orbited by another, inside a rotating machine, in an eddy advected through a shear — the criteria can be confidently wrong.

Measuring the fate directly #

The fluid’s fate can be measured without any frame at all, by following trajectories and asking how fast neighbours separate. That is the finite-time Lyapunov exponent, and it is the same for every observer. For one flow from each region of the map it does exactly what the rotating-frame analysis predicts. Haller’s false vortex settles at one; a steady saddle with a little vorticity, at 0.87, which is $\sqrt{1 - 1/4}$; a steady vortex and a hidden vortex both fall towards zero, as their particles circulate and return. The exponent does not care what any snapshot said.

This is why the objective definitions of a vortex that where a vortex stops describes are built on trajectories over an interval, and why the boundary that only exists over a window is drawn from the Lyapunov exponent rather than from the velocity field. Haller’s flow is the reason those definitions exist: it is the smallest example of a snapshot criterion failing in the laboratory frame, and it was constructed to make that point.

What two snapshots can say that one cannot #

The missing quantity is not mysterious; it is the rate at which the strain’s axes turn, and that is a time derivative. Two snapshots a short interval apart give it, and with it the corrected condition $|\omega - 2\Omega_s| > 2s$ is a local, nearly instantaneous criterion. This correction was proposed for two-dimensional turbulence by Lapeyre, Klein and Hua in 1999, as an “effective rotation” that adds the turning of the strain’s axes to the vorticity, and it sharpens the Okubo–Weiss partition exactly where the uncorrected version fails.

It is still not the fluid’s fate. The corrected criterion is right for a strain turning at a steady rate, which is the family here; a strain whose turning rate itself changes, or a particle that moves into a region of different strain, needs the history over the whole interval of interest, and only a trajectory integral supplies that. What the correction shows is that the gap between a snapshot and a trajectory is not all-or-nothing. The first term of the gap is the strain’s rotation, it is computable from the velocity field and its rate of change, and for many flows it is most of the error.

Where rotating strain happens in real flows #

The family is linear and uniform, but the situation it models is common. The strain a vortex feels is set by everything else in the flow, and when the rest of the flow moves, the strain’s axes turn. Two co-rotating vortices move each other in a circle, and each sits in a strain from its partner whose axes turn with the pair. Between them, and in the fluid just outside each core, the strain’s rotation is comparable with the vorticity, which puts those regions near the wedges of the map: an instantaneous criterion drawn there can report a bridge of vortex between the two cores that the fluid does not actually share.

The same arithmetic applies to two strainings applied in sequence, where the order of the strains matters because their axes differ, and to the stretching rate that is not one number, where a material line’s growth depends on how long it stays aligned with a stretching direction that may be moving. In each case what decides the outcome is a history, and a snapshot holds none.

The largest-scale use of the snapshot criteria is in the ocean, where satellite altimetry gives the surface velocity field every day and eddies are counted by the Okubo–Weiss parameter — the two-dimensional Q. The eddies that matter carry heat, salt and plankton across ocean basins, and whether a detected eddy actually carries its water depends on exactly the distinction drawn here. In quiet regions the strain round an eddy turns slowly and the census is sound. In the energetic regions where eddies are shed and merge — along western boundary currents, where they are most numerous — neighbouring eddies turn one another’s strain at rates comparable with their own vorticity, and a census taken from snapshots counts false vortices and misses hidden ones in the proportions this map implies. Comparisons with Lagrangian eddy boundaries have found snapshot contours enclosing water that leaks out within days, which is the observational form of Haller’s example.

What the picture cannot show #

A linear flow. Every point of the family has the same gradient, so the flow has no boundary and no core. A real vortex’s gradient varies with position, and a region can be a false vortex in its outskirts while its core is genuine.

A strain that turns steadily. The axes turn at a constant rate for ever. In a real flow the turning rate changes, and the map says what happens at each rate, not what happens when the rate itself varies.

Two dimensions. In three dimensions the criteria and the stability analysis both acquire a third direction, and the wedges become regions of a larger space; the principle — that the criterion reads the gradient and the fluid obeys the history — is unchanged.

The convention the numbers depend on #

The strain rate $s$ is the positive eigenvalue of the rate-of-strain tensor. The vorticity $\omega$ is twice the rotation rate and is negative for clockwise rotation, as in Haller’s example. $\Omega_s$ is the rate at which the strain’s principal axes turn, also negative for clockwise. Q is the second invariant of the velocity gradient, $\omega^2/4 - s^2$ in two dimensions. Times are in units of one over the strain rate. The finite-time Lyapunov exponent is the logarithm of the flow map’s largest singular value over the interval, divided by the interval’s length, and it is the same for every observer.

Who found it, and when #

George Haller published the rotating saddle in 2005, in a paper proposing an objective definition of a vortex, as the demonstration that instantaneous criteria can fail in the frame in which a flow is measured. The stabilisation of a saddle by rotation, the other pair of wedges, is much older in mechanics: it is the principle of the rotating saddle trap and, in electric fields, of the Paul trap, for which Wolfgang Paul shared the 1989 Nobel Prize in physics. The Q-criterion is Hunt, Wray and Moin’s of 1988, and $\lambda_2$ Jeong and Hussain’s of 1995.

Still open: a pair that orbits #

The map is drawn for a uniform gradient. The flow near two co-rotating point vortices is not uniform, but its strain turns at a known rate, the pair’s orbital frequency, and its vorticity and strain vary smoothly between the cores. The calculation that follows computes, at every point around such a pair, the snapshot criterion and the finite-time Lyapunov exponent over one orbit, and marks where they disagree — which should be a region of false vortex between the cores and hidden vortex outside them, placed where this map says. It would say how much of the “vortex” a Q-criterion draws round an orbiting pair actually travels with it, which is the quantity the earlier essay’s two observers disagreed about by a factor of twelve.