Where a vortex stops

Every essay on this site that says the vortex has been relying on a reader to see one. For a point vortex that is fine, because a point vortex has no size. It stops being fine the moment a figure shades a region and calls it the core, because the shading needs a boundary and the equations do not supply one.

Vorticity is a field. A vortex is an object. Turning the first into the second is a decision, and this essay is about who makes it.

Four criteria, and three of them are one #

The literature offers four local tests. Written for the velocity gradient $\nabla\mathbf u$ decomposed into its symmetric part $\mathsf D$ and antisymmetric part $\mathsf W$:

$$Q = \tfrac12\big(|\mathsf W|^2 - |\mathsf D|^2\big),\qquad \Delta = \mathrm{tr}^2 - 4\det,\qquad \lambda_2\ \text{of}\ \mathsf D^2 + \mathsf W^2,$$

with a vortex where $Q > 0$, or where $\Delta < 0$, or where $\lambda_2 < 0$. They are introduced as three ideas: rotation beating strain, eigenvalues going complex, a pressure minimum in the plane perpendicular to the vortex.

In two dimensions, incompressibly, they are one idea. For a trace-free gradient $[[a,b],[c,-a]]$,

$$Q = -(a^2 + bc) = \det\nabla\mathbf u\quad\text{exactly},\qquad \Delta = -4Q,$$

so all three change sign on one curve — measured at $10^{-16}$ over sixty sample points on both sides of it. And that curve is the boundary between the points where the local flow pattern is a focus or a centre and the points where it is a saddle, which is exactly the classification topology applies to stagnation points, applied at every point instead of only at the special ones.

That is worth knowing before choosing between them. A paper comparing $Q$ and $\lambda_2$ on two-dimensional data is comparing a quantity with itself.

The one that really is different #

The vorticity magnitude is not one of them, and the cleanest demonstration is the flow this collection uses for every stability argument.

A parallel shear layer, $u = \Delta U\tanh(y/h)$, has vorticity peaking at the centre and falling away on both sides. It has $Q = 0$ at every point, because in a parallel shear the strain and the rotation have exactly equal magnitude — the velocity gradient has one non-zero entry, which splits evenly between the two parts.

So a vorticity threshold calls the entire layer a vortex and $Q$ calls none of it one.

Neither is a mistake. They are answers to different questions, and the question a reader usually has in mind — is there a swirling structure here? — is the second one. The habit of contouring vorticity and calling the result the vortices is what makes a shear layer look, in a great many published figures, like a row of vortices before it has rolled up into any.

Where one vortex ends #

Now the case that has a real answer. A Lamb–Oseen vortex — the one everybody means by a vortex with a core — has a Gaussian vorticity distribution and a swirl that rises linearly, peaks, and falls off as $1/r$.

Its vorticity is positive at every radius, so a vorticity threshold puts the edge wherever the threshold is put. Its $Q$ changes sign exactly once, at

$$r = 1.1209\,r_c,$$

and that radius encloses $71.53$ per cent of the circulation. It is a property of the flow. Nothing was chosen.

That is as good as it gets, and it is worth pausing on how good it is: a definite radius, a definite fraction of the circulation, and no free parameter. The rest of this essay is about the two ways it is usually thrown away.

The first way: a level #

Almost nobody plots $Q > 0$. What gets plotted is $Q > c$ for some $c$, because the $Q > 0$ set in a turbulent field is enormous and connected and shows nothing.

The consequence is arithmetic. Sweeping $c$ from zero to seven-tenths of $Q$’s peak takes the radius of the region from $1.121\,r_c$ down to $0.416$, and the enclosed circulation from $71.5$ per cent to $15.9$.

There is no principled way to pick $c$. It is usually chosen so that the picture looks like what the author expects, and the resulting figures are then compared between papers that chose differently. The honest presentation is to state the level and the fraction of circulation it encloses, and this collection now does.

The second way: a frame #

The deeper problem is that $Q$ is not objective. The velocity gradient’s antisymmetric part shifts by the observer’s own rotation rate, so $Q$ shifts too, and where the vortices are becomes a question with a camera in it.

Take two Lamb–Oseen vortices side by side. They orbit their common midpoint, so the natural frame in which to look at them is the one that turns with the pair — which is exactly what an analyst does with a rotor, a stirred tank or a vortex pair.

In the laboratory the $Q > 0$ set is two compact cores, of total area 2.00 in the window drawn. In the co-rotating frame it is those two cores plus everything beyond a certain distance, of total area 24.94 — because fluid that is nearly at rest in the laboratory is going round in a rotating frame, and $Q$ cannot tell the difference between that and a vortex.

The far field is not a vortex. Every criterion in the list says it is, to an observer who is turning.

Why the collapse happens, and what it means #

The identity $Q = \det\nabla\mathbf u$ is worth an extra minute, because it explains why the three criteria were ever thought to be different.

In three dimensions the characteristic polynomial of the velocity gradient has three invariants — $P$, $Q$ and $R$ — and for an incompressible flow $P$ vanishes, leaving two. $Q$ is the second of them and $\det\nabla\mathbf u$ is the third, so they are genuinely independent quantities and the criteria built from them are genuinely different tests. The whole $Q$–$R$ plane, with its famous teardrop of turbulence statistics, exists because the two are independent.

In two dimensions there is only one non-trivial invariant, so every scalar built from a trace-free $2\times2$ matrix that is invariant under rotation must be a function of it. Three tests that differ in three dimensions therefore have no room to differ in two. That is not a coincidence being explained away; it is a statement about how much information a plane velocity gradient contains, which is three numbers after the trace is removed and one number after rotations are quotiented out.

The practical consequence: a two-dimensional study cannot distinguish between the criteria and should not claim to, and a three-dimensional study that finds them agreeing has learned something rather than nothing.

What a reader is usually asking #

The criteria answer a question about $\nabla\mathbf u$ at a point. Almost nobody wants that answer.

What a reader looking at a vortex figure usually wants to know is one of:

Which fluid stays together? That is a question about trajectories over an interval, and the answer is a material region — the objects advection is about. It has no pointwise definition at all: whether two parcels stay together depends on how long is meant by stay.

Where would a tracer accumulate? In an incompressible flow, nowhere — a material region keeps its area — so the question is really about where a tracer appears to accumulate over the time somebody is watching, which again has an interval in it.

What carries the circulation? This one has a good answer and it is the one this essay recommends quoting: the fraction of the total circulation inside a stated contour. It is a conserved quantity, it is frame-independent under a boost, and it is far less sensitive to the level than the area is.

And what will happen next? Which no diagnostic answers, because a diagnostic is a function of the instantaneous field and the future is not.

The thing that does not move #

There is one quantity in all of this that survives every objection, and it is the one this collection has been using since its second essay.

Circulation. It is an integral of vorticity over an area, it is unchanged by a Galilean boost, it is conserved for a material loop in an inviscid fluid, and it needs no threshold and no criterion — only a loop, which the analyst still chooses but whose choice is visible in the answer rather than hidden in it.

That is why every result in this collection about vortices is stated in terms of circulation and not in terms of a core: the lift on a wing, the speed of a vortex pair, the induced drag of a wake are all $\Gamma$-statements. The core radius appears in exactly one place — the logarithm in a vortex ring’s self-induced speed — and that essay is careful to assert the slope of the dependence rather than the value, for precisely the reason this one is about.

A worked disagreement #

To make the level’s cost concrete, here are two entirely reasonable analysts looking at the same Lamb–Oseen vortex.

The first plots $Q > 0$ and reports a vortex of radius $1.121\,r_c$ containing $71.5$ per cent of the circulation and an area of $3.95\,r_c^2$.

The second plots $Q$ greater than a fifth of its peak — a common choice, made because it renders well — and reports a vortex of radius $0.832\,r_c$ containing $50.0$ per cent of the circulation and an area of $2.17\,r_c^2$.

Neither has done anything wrong. Their radii differ by thirty-five per cent, their areas by a factor of 1.8, and their circulations by a factor of 1.4. If the two numbers being compared were a measured core size and a theoretical one, the comparison would be reported as a discrepancy and somebody would go looking for the physics in it.

What is not in the velocity field #

Three things, and the essay is the list.

A threshold, which nothing supplies and which moves the reported size and strength over most of their range.

A frame, which nothing supplies and which can multiply the reported area by twelve.

And an idea of what a vortex is for. The criteria answer is the local flow pattern closed, which is a question about $\nabla\mathbf u$ at a point. A reader usually wants to know something else: which fluid stays together, where a tracer accumulates, what a following observer would see as coherent. Those are questions about trajectories over an interval, and no pointwise criterion answers them.

That last is not a complaint about the criteria; it is a statement about what they are. Material lines stop being pulled apart exactly on the $Q = 0$ surface — which is the best justification $Q$ has and is derived rather than asserted — but the derivation is for the rotation of the strain axes relative to the material, and an observer’s own rotation is added to it in every published $Q$ field.

The criteria that do not have a frame in them #

The two objections in this essay — a threshold nobody supplies and a frame nobody states — are not unanswerable. There is a family of definitions built to remove both, and what they cost is exactly the thing the essay has been circling: they are not diagnostics of a snapshot.

The construction begins by noticing why $Q$ moves with the observer. An observer’s own rotation adds the same rigid amount to the vorticity everywhere, so a quantity built from the deviation of the vorticity from its instantaneous spatial mean cannot see it. Average that deviation along a fluid trajectory over an interval, and the result is a scalar attached to each parcel that is the same for every observer, however they are turning.

A vortex is then defined as a region of large values of that scalar, and the threshold problem is attacked separately: rather than choosing a level, take the outermost level set that is still convex around each local maximum. Convexity is the discriminating condition, because a boundary that has begun to filament — to be drawn out into the surrounding strain — stops being convex, and filamenting is precisely what a coherent vortex boundary is defined as not doing.

What comes out is a boundary that is objective, nearly threshold-free, and material: it is made of the same fluid at the end of the interval as at the beginning. Applied to ocean eddies, boundaries of this kind retain their water for months, where the contours drawn from instantaneous fields leak steadily — which is the practical form of the essay’s complaint, since the whole point of calling an eddy an object is that it carries something along.

And the price is the interval. These definitions need trajectories over a stated span of time, so they are not functions of the velocity field at one instant, and the answer depends on how long “coherent” is taken to mean. That is a free parameter, exactly as the level was — with the difference that it is a physically meaningful one, chosen by the question being asked, and visible in the answer rather than hidden inside a colour scale.

Which is the honest resolution rather than an escape. There is no snapshot diagnostic of a material object, and the criteria in this essay are snapshot diagnostics.

What an honest figure does #

Four habits, and this collection now keeps all four.

Name the criterion. A vortex is not a measurement. The region where $Q$ exceeds five per cent of its peak, in the laboratory frame is.

Name the level and what it encloses. The fraction of circulation inside a contour is a far more stable number than the contour’s area, and it is the one a reader can compare across papers.

Name the frame, and say whether the answer moves if it changes. For an isolated vortex in a stream it does not much; for anything rotating it does.

And prefer $Q = 0$ when the flow allows it. For a single core it is a real boundary with a real enclosed circulation, and it is the only level in the family that is a property of the flow rather than of the author.

What the criteria are actually for #

None of this makes them useless, and it is worth saying what they are good at.

They are excellent at finding structures in a large data set — as a detector, where the threshold’s arbitrariness costs nothing because the question is where to look rather than how big something is. They are good at counting, provided the count is robust to the threshold, which is a thing that can be checked and rarely is. And they are bad at measuring, because every measurement they produce moves with the level.

That maps onto the same distinction this collection has drawn about what a photograph of a flow shows: a visualisation is evidence that something is there and is very poor evidence about how large it is. The criteria are a numerical visualisation and inherit exactly that property.

The model limit #

Two, and the second is the one that matters at scale.

Everything computed here is two-dimensional, and the collapse of three criteria into one is a two-dimensional result. In three dimensions $Q$, $\Delta$ and $\lambda_2$ are genuinely different functions and identify genuinely different sets — usually similar, occasionally not, and the cases where they differ are the interesting ones. The vorticity-threshold objection survives unchanged.

There is also a quiet assumption in the whole framework that the flow is incompressible, and it is doing work. The trace of the velocity gradient is what was set to zero to make $Q$ equal the determinant, and a compressible flow has a divergence which enters every one of these quantities. In a shock-containing flow the criteria pick out the shock as enthusiastically as they pick out a vortex, and the usual fix is to subtract the dilatational part first — which is the decomposition of the previous essay with all of its own ambiguity attached.

And the Lamb–Oseen vortex used throughout is an isolated, axisymmetric, steady structure, which is the easiest possible case. A vortex in a turbulent flow is none of those things: it is being strained by its neighbours, it is not axisymmetric, and it is changing while it is being measured. The $Q = 0$ radius that is a clean property of the flow here is, there, a surface that moves while the measurement is being made — and the level, the frame and the averaging interval all come back.