Part II · put the representation to work

Follow the labels.
See a velocity emerge.

Clebsch notation can look like a trick because three innocent scalar fields suddenly become a velocity. This page slows the trick down. First watch two labels travel with the water; then ask what a one-form measures; finally assemble one small field by hand.

\eta=d\phi+\alpha d\beta,\qquad \omega^{(2)}=d\eta=d\alpha\wedge d\beta,\qquad \zeta=\star_g\omega^{(2)}

You do not need to memorize the formula before playing. The experiments are arranged so that each term earns its meaning.

First experiment · Clebsch variables as evolving state

Carry the labels.
Let φ change.

PERIODIC TAYLOR–GREEN FLOW · ZERO HARMONIC PERIODSdeforming material-label grid

α contours β contours tracked patch

SHOW

Follow the orange tracer

The representation evolves even though the Euler velocity is steady.

\alpha_0=2U\cos x_0,\qquad\beta_0=\cos y_0,\qquad\phi_0=-U\cos x_0\cos y_0

01
α and β stay attached. Cyan and orange curves are connected pieces of material level sets. Their shapes deform, but the tracer’s two label values remain constant. The pair labels fluid parcels; it need not be a one-to-one coordinate chart.
02
Their crossing carries a vorticity two-form. Intrinsically, $\omega^{(2)}=d\alpha\wedge d\beta$ . The colored background is the scalar $\zeta=\star_g\omega^{(2)}=2U\sin x\sin y$ . If the surface is embedded and oriented, the familiar normal vorticity vector is $\boldsymbol\omega=\zeta\,\mathbf n$ .
03
φ absorbs the pressure contribution. Its tracer value evolves by the Bernoulli equation above. This is how pressure can act while α and β remain material invariants.

TIME0.000

CARRIED LABELSα — · β —Δα = Δβ = 0

EXACT POTENTIALφ —Δφ —

INCOMPRESSIBILITY AUDITarea ratio —Δω —

This Taylor–Green comparison has zero harmonic periods: there is local vortex motion but no uniform drift around either torus cycle. The translucent quadrilateral is a small material patch; its area should remain one times its initial area. This controlled steady-flow test isolates the Clebsch transport laws; it is not yet a free-running nonlinear solver. When stretching outruns the contour sampling, the renderer leaves a small gap instead of inventing a straight crossing.

NEXT QUESTION · WHAT DID THIS ZERO-PERIOD TEST LEAVE OUT? Add the two harmonic circulations on the flat torus → Keep the local vortex pattern while changing global particle winding around the x and y cycles.

Second experiment · Chapter 1 picture · dual pairing

Make the covector visible.

Before assembling a velocity one-form, see what a one-form actually does. Fix $\eta=s\,dx$ , rotate the orange tangent vector, and watch the pairing count signed crossings. The cyan arrow is $\eta^\sharp$ ; it appears only after choosing the Euclidean metric.

covector level linesmetric-dual η♯test vector v

THE NUMBER RETURNED BY THE COVECTOR

\eta(v)=1.2\cdot1.0\cos 55^\circ

η(v) = 0.688

The vector crosses the level lines in the positive direction.

Angle of the tangent vector v90° follows a level line; 180° reverses direction55° Length of va longer displacement can cross more level lines1.00 Covector strength slarger s packs the level lines more densely1.20

WITHOUT A METRICThe pairing already exists.

$\eta_p(v)$ only needs a vector and a covector at the same point.

WITH A METRICWe may draw η as an arrow.

The metric defines $\eta^\sharp$ , so in this Euclidean picture $\eta(v)=g(\eta^\sharp,v)$ .

ALONG A FLUID PATHAdd the local readings.

Set $v=\dot\gamma(t)$ and integrate: $\int_\gamma u^\flat$ is circulation.

How the potentials are chosen

Do not choose three mysterious fields at once.

Start from the physical velocity you want. Split its specification into local vorticity and, when the surface has handles, global loop circulations. The Clebsch fields are a nonunique factorization of that data—not three additional observables.

01
Name what is local and what is global. $\omega=du^\flat,\qquad \Gamma_i=\oint_{\gamma_i}u^\flat$
The two-form is local circulation density. The periods are additional state only for noncontractible cycles; a genus-g surface has 2g of them.
02
Choose two label families whose crossing matches it. $d\alpha\wedge d\beta=\omega$
This factorization is local and highly nonunique. In a material formulation, these labels can instead be initialized once and transported with the fluid.
03
Complete the velocity, including its periods. $u^\flat=d\phi+\alpha d\beta+h,\qquad h\in\mathcal H^1(S)$
Locally the closed remainder is dφ. Globally, h supplies any periods the chosen labels do not carry. An incompressible projection changes only the exact part.

Third experiment · a small construction game

Build a swirl from two painted labels.

01
Lay the labels parallel. Why is there no swirl yet?
02
Let them cross. Watch oriented area—and vorticity—appear.
03
Add and remove an exact piece. Change velocity without changing its curl.

A TWO-MINUTE PATH THROUGH THE CONTROLSUse the presets as a story, not a test.

Move left to right. Keep the vorticity colors visible and ask what changes after each click.

Choose a surface

Choose a field case

Build the one-form

A NOTE ABOUT FREEDOM The labels are a description, not three new observables.

Many triples describe the same velocity. Here we choose especially simple sine waves so you can see what each term does; a solver may choose labels for numerical convenience, transport, or structure preservation.

YOUR CURRENT FIELDperiodic plane · two crossing label foliations

metric-dual arrows selected probe

drag to orbit · click the surface to move the probe

color: α · arrows: (dα)♯

STEP 01 · LABELS

\alpha,\beta\in\Omega^0(S)

SMOOTH VIEW: evaluate at a point · MESH VIEW: sample at vertices

IN THE PERIODIC-PLANE EXAMPLE

\alpha=A\sin x,\quad\beta=\cos\theta\sin x+\sin\theta\sin y

The angle θ controls whether the two label families change in the same direction or independently.

Build one controlled example

Label crossing0° parallel → 90° independent70° Label strengthscales α and its vorticity0.70 Potential strengthchanges dφ, never dα∧dβ0.24

WHAT THE CURRENT CHOICE MEANS

\alpha=0.70\sin x,\quad\beta=0.34\sin x+0.94\sin y

\omega^{(2)}=d\alpha\wedge d\beta=0.66\cos x\cos y\,dx\wedge dy

\phi=0.24\cos(x-y),\qquad d(d\phi)=0

The sine fields are only a simple periodic example. Their crossing determines vorticity; φ independently changes the curl-free part of velocity.

Choose what to inspect

Glyphs show metric duals

POINT PROBEu = 0.19 · v = 0.31

α—

β—

φ—

ζ = ⋆ω²—

At this point, the two label differentials meet at an angle and contribute an oriented-area two-form. The number above is its Hodge-dual scalar.

Background notebook · where this picture sits

Clebsch variables are one member of a larger family of fluid descriptions.

This playground teaches a shared geometric language; it does not claim that every related method runs the same algorithm. The useful comparison is to ask what each method stores, what it transports, and which conservation law motivates the choice.

LABELSClebsch maps
Use transported scalar structure to expose and process vortices. This is the closest relative of the colored label-grid picture above.
CIRCULATIONCovector fluids
Reformulate velocity advection so that the numerical update respects the one-form quantity measured along moving loops.
TOPOLOGYFluid cohomology
Add the harmonic periods that local vorticity cannot recover on domains with holes, periodic boundaries, or genus.
DYNAMICSVakonomic fluids
Use low-rank Clebsch-like variables inside a particular structure-preserving discretization of Euler flow. That is a research destination, not what this local construction demo implements.

A good place to pause

The local picture is now complete.

You can choose scalar labels, differentiate them into one-forms, assemble $u^\flat=d\phi+\alpha\,d\beta$ , and use the metric only when an arrow is needed. The next surprise is global: a loop around a hole can remember circulation even when local vorticity says nothing.

Follow the labels.See a velocity emerge.

The local picture is now complete.

Follow the labels.
See a velocity emerge.