An edge no observer can move still depends on what it is averaged against

Where a vortex stops took two equal Lamb–Oseen vortices side by side and drew where the Q-criterion puts them. In the laboratory the set where $Q$ is positive is two compact cores, of total area 2.00 in a window of 46.2. In the frame that turns with the pair it is 24.94 — the cores and nearly all the fluid around them — because the antisymmetric part of the velocity gradient shifts with the observer’s own rotation, and $Q$ cannot tell the far field’s apparent swirl from a vortex. The same vorticity field, two observers, a twelvefold difference in area — and every snapshot says vortex had already shown that a frame which makes a flow look like a vortex can do so while every parcel in it leaves.

That essay ended by describing the cure without computing it. An observer’s rotation adds the same amount, $2\Omega$, to the vorticity everywhere, so the vorticity’s deviation from its spatial mean is the same to every observer. Average that deviation along each fluid parcel’s path over an interval and the result is a number attached to each parcel that no rotation of the observer can change. Then call the vortex the region inside the outermost level of that number that is still convex round its peak, because a boundary that has begun to be drawn out into filaments is not convex, and a vortex is defined as a region that is not being drawn out.

This essay computes it for the same pair, and finds that it does what was claimed for it — and that it has a setting of its own, which the description did not mention.

The pair, and the number carried along each path #

The two vortices have circulation $2\pi$ each and core radius 0.5, and sit 3 apart. Two equal point vortices that distance apart orbit their midpoint at $\Omega = \Gamma/\pi d^2 = 0.2222$, one orbit in 28.27 time units, and here the cores orbit rigidly at that rate without deforming. That is a prescribed flow rather than a solution of Euler’s equations — a real pair’s cores are slightly elliptical, and with viscosity they spread and eventually merge — but it is the flow the earlier essay drew, and the question here is about a definition, not about the pair’s dynamics.

In the laboratory the flow is unsteady: the cores go round. In the frame turning at $\Omega$ it is steady, the velocity is the laboratory’s minus $\boldsymbol\Omega \times \mathbf x$, and the vorticity is the laboratory’s minus $2\Omega$ everywhere. Each observer computes, for a parcel starting at $\mathbf x_0$,

$$\mathrm{LAVD}(\mathbf x_0) = \int_0^T \big|\,\omega(\mathbf x(t), t) - \bar\omega(t)\,\big|\;dt,$$

the Lagrangian-averaged vorticity deviation, with $\bar\omega$ the mean vorticity over a region. The region here is a disc of radius 6 about the orbit’s centre, which turns into itself as the pair goes round, so both observers agree on it and its mean is constant in time: 0.1111 in the laboratory, $0.1111 - 2\Omega$ in the turning frame. The paths are integrated by fourth-order Runge–Kutta.

Computed separately by the two observers, on a grid of 101 by 101 starting points over the window and one orbit, the two fields differ nowhere by more than $8 \times 10^{-8}$ of their peak. That is the whole of the objectivity claim, and it holds to integration error.

A disc twice the core’s radius #

The field has a sharp peak at each core, where a parcel spends the whole orbit in vorticity far above the mean, and its levels round each peak are nearly circles. The outermost of them that is still convex is at 1.77, and it encloses an area of 3.15 round each core: a disc of radius 1.00, twice the core radius. The pair’s two edges together hold 6.29 of the window, three times what the laboratory’s $Q$ called vortex and a quarter of what the turning frame’s $Q$ called vortex.

That puts the edge somewhere a reader might have guessed. A Lamb–Oseen vortex’s vorticity at twice its core radius is $e^{-4}$ of its peak, under two per cent, and the circulation inside that radius — the vorticity added up over the disc — is 98 per cent of the total. The edge is the vortex with nearly all of its circulation and none of the irrotational fluid swirling round it — which is the part $Q$ in a rotating frame mistook for vortex.

Why the contour stops where it does #

The construction’s last step is less dramatic than its description. Scanning the levels down from the peak, the closed contour round a core grows as a near-circle whose convexity deficiency — the share by which its convex hull’s area exceeds its own — stays below 0.0054 all the way to an area of 3.15. Then it does not lose convexity gradually. It stops closing round the core altogether.

The reason is the far field. A parcel far from both cores sits in vorticity near zero, so its deviation from the mean is the mean itself, and over one orbit it accumulates $0.1111 \times 28.27 = 3.14$. Between the cores’ peaks and that plateau the field has a trough, a ring round each core where the core’s own vorticity has fallen to the mean and the deviation passes through zero. Below the trough’s level a contour round a core has nowhere to close: it runs out to the far field. The edge is therefore the trough’s rim, and the trough sits where a core’s vorticity equals the disc’s mean — at a radius of 1.03 for a disc of radius 6.

That observation is what makes the next figure necessary.

The setting nobody mentioned #

The mean is a mean over something. Both observers agree on a disc about the orbit’s centre, which is what keeps the measure objective, but nothing in the construction says how large the disc should be — and the trough, and with it the edge, sits where a core’s vorticity falls to whatever that mean is.

A disc of radius 4 has a mean of 0.25, and the edge comes in to 0.896. A disc of 6, the one used above, gives 1.00. A disc of 10 has a mean of 0.04 and the edge moves out to 1.13; a disc of 20, with a mean of 0.01, leaves it at 1.13. On small discs the edge is the trough, and follows the mean. On large ones the trough has moved out past the point where the contours remain convex — at 1.29 for the disc of 20 — and the edge is set by the convexity tolerance instead, so it stops moving. Between the smallest and the largest disc the edge’s radius changes by 26 per cent and its area by more than half.

That is the refuted claim’s gap. The construction is objective and nearly free of its tolerance, both of which are true and were checked. It is not free of choices: it trades the Q-criterion’s level and frame for an averaging region and a window. The window turns out to be nearly harmless, as the figure after next shows. The region does not.

There is a principled answer, which is to take the region that contains the flow being studied and no more: in an ocean basin, the basin; in a computation, the domain. That is defensible, and it is still a choice. For an isolated pair in an unbounded fluid there is no such region, the mean tends to zero as the disc grows, and the edge settles at the convexity limit, 1.13. Whether that or the trough at 1.00 is “the” edge is a question the definition cannot answer by itself.

Which edge keeps its fluid #

A coherent vortex is supposed to carry its fluid with it, so there is an independent test that involves no averaging at all: release fluid within a given radius of a core and see whether any of it leaves.

The test follows parcels in the laboratory and asks whether any wanders more than 1.5 — half the separation, the distance to the saddle between the cores — from the moving centre of the core it started round. From within a radius of 1.00, none does. The largest radius with no loss, found by bisection, is 1.03. Beyond it the loss starts at once and grows fast: 1.6 per cent of the fluid from within 1.05, 19.8 per cent from within 1.2, 47.1 per cent from within 1.4. Three orbits lose exactly the fluid that one orbit does, so the losses happen in the first orbit and the remainder is held indefinitely, at least in a pair that never deforms.

So the disc of radius 6 gives an edge at 1.00 that keeps all its fluid, and the trough at 1.03 sits exactly at the retention boundary — the vorticity at the mean and the last circle of fluid that stays are the same circle to within the bisection’s step. The large-disc edge at 1.13 holds fluid of which a tenth, 9.9 per cent, leaves its core within the first orbit. The small-disc edge at 0.896 is conservative.

Two things follow. The averaging region is not a free cosmetic choice: one setting draws a boundary that leaks and another draws one that does not. And the retention boundary itself is not an LAVD quantity at all; it is a direct statement about where fluid goes, computable whenever the paths are. The LAVD edge is valuable because it finds a near-retention boundary from one field without running a test for every candidate radius, and it is accurate here when the region is chosen to make the mean what the far field actually is.

The window hardly matters #

The other choice the construction needs, the interval over which paths are followed, turns out to matter very little for this flow. An edge computed over a quarter of an orbit has radius 0.991; over half an orbit, 0.998; over one, two or three orbits, 1.00 within a part in five hundred. The convexity tolerance matters only when it is set very tight: 0.939 at a tolerance of 0.001, 0.961 at 0.002, and 1.00 for every tolerance from 0.01 to 0.05, because above about 0.005 the edge is the trough’s rim rather than a convexity limit.

The window is harmless here because the pair is steady in its own frame: a parcel’s history over a quarter orbit is a fair sample of its history over three. A boundary that only exists over a window found the opposite for a flow that is genuinely unsteady, where the ridge of the stretching field moved with the interval and reversed with time’s direction. The LAVD edge inherits the window from the same Lagrangian construction, and in an unsteady flow it would inherit the same dependence.

Against the eddies the pair carries #

In the turning frame the pair’s streamlines have a structure of their own: a saddle between the cores, two more beyond them, and the separatrices joining them enclose two large eddies no snapshot can see, fluid that turns with the pair without carrying any vorticity. The LAVD edges sit well inside the separatrix lobes round each core and say nothing about those eddies. That is correct by construction: the eddies’ fluid has vorticity near zero, its deviation is the mean’s, and the LAVD of a parcel that turns with the pair forever is the same as that of a parcel at rest far away. A measure built from vorticity cannot find a structure made of irrotational fluid, however coherent the structure is — the same distinction spin is not going round draws between a fluid’s local rotation and its orbit. The turning frame’s $Q$ found too much; the LAVD finds the cores and only the cores; the eddies need a third tool — the frame’s own streamlines, which are not objective, or a retention test like the one above, which is.

What a reader of a vortex census should ask #

The practical consequence concerns any count of coherent vortices built this way — ocean eddies from satellite altimetry, vortices in a simulation. The census is objective: it does not depend on how the analyst’s frame turns. It is nearly free of the convexity tolerance and, for slowly evolving structures, of the window. It depends on the averaging region, and the region is rarely stated with the same prominence as the window. Two censuses of the same eddies with different regions draw edges whose radii can differ by a quarter and whose areas — and therefore the volume of water each eddy is said to carry — by half. The number to ask for is the region, and the check to ask for is a retention test like the one here.

The convention: what each number is #

Lengths are in the units of the pair’s separation divided by three, so the cores’ radius is 0.5 and their centres are 3 apart; times are in the same units with circulation $2\pi$, so an orbit is 28.27. The LAVD is accumulated by the trapezium rule along fourth-order Runge–Kutta paths with a step of 0.02. Contours are taken from the field on a grid of 101 by 101 over a square of half-width 3.4, the earlier essay’s window, and scanned at logarithmically spaced levels. The convexity deficiency is the convex hull’s area minus the contour’s, over the contour’s. Areas and radii are per core unless stated; the radius quoted is that of the disc with the contour’s area.

What the model leaves out #

Deforming cores. The cores orbit rigidly; in a real pair they are slightly elliptical and nutate, which adds a weak unsteadiness in the turning frame and makes the window matter a little. Viscosity, which spreads the cores, moves the trough outward with time and ends in merger. Unequal vortices, whose orbit centre is not the midpoint and whose disc of agreement is therefore different. Three dimensions, where the deviation is of a vector and the convexity condition has to be replaced by something about surfaces. And the averaging region’s shape: a square or a basin-shaped region would give a different mean and a different edge again, which is the essay’s point carried one step further rather than a separate caveat.

Who devised it #

The Lagrangian-averaged vorticity deviation and its outermost-convex-level boundary are due to Haller, Hadjighasem, Farazmand and Huhn, in 2016, who called the resulting regions rotationally coherent vortices and applied them to eddies in ocean surface velocities. Their definition takes the spatial mean over the domain of the data, which is the principled choice the essay describes; the dependence on that domain is a property of the definition rather than a criticism of its use.

How it was checked #

The two observers’ LAVD at five parcels — in a core, near the central saddle, in the far field, and between — agree to $1.1 \times 10^{-7}$ of the value, and the whole fields to $8 \times 10^{-8}$ of their peak. Halving the step moves a parcel’s LAVD by four parts in ten thousand. The disc’s mean vorticity matches the two cores’ circulation over the disc’s area to two parts in ten thousand. On a grid of 61 by 61 the edge’s radius is 0.980, against 1.00 on 101 by 101. The retention test’s zero loss inside 1.00 holds on eight rings of parcels as well as on twenty-four, and the loss from within 1.3 stays above a fifth on both. The construction refuses an unknown observer, a window that is empty or negative, and an averaging disc that does not contain the cores’ orbit.

Still open: a pair that merges #

Every number here is for a pair whose cores never change. Two real vortices close enough together — a separation of a few core radii, against the six here — merge: their orbit speeds up, the cores are drawn out, and filaments are shed as one vortex forms. That is exactly the situation a vortex boundary is needed for and the one in which the window cannot be harmless. The next calculation would follow a merging pair of finite vortex patches with contour dynamics, compute the LAVD edge over windows ending before, during and after the merger, and ask whether the fluid inside the edges before merger is the fluid inside the merged vortex’s edge afterwards — and how much of what the edges excluded the merged vortex nevertheless swallows.