Begin with the physical problem

A fluid must
move and remember.

A surface-fluid solver must move particles and remember how the velocity acts around loops. Before introducing Clebsch variables, separate two words that are easy to blur: circulation is attached to a loop; vorticity is attached to an area.

\Gamma(C)=\oint_C u^\flat,\qquad \omega=d(u^\flat),\qquad \int_A\omega=\oint_{\partial A}u^\flat

The solver ultimately needs the velocity vector $u$ . Clebsch variables are one possible representation of its metric-dual one-form $u^\flat$ ; they are not a new kind of fluid.

One idea at a time

Start with one rotating flow.

We will describe exactly the same counterclockwise motion four ways. The objects are related by translation; they are not four successive simulation steps.

1D · LOOP MEASUREMENT

Circulation asks about one chosen loop.

\Gamma(C)=\oint_C\eta,\qquad \eta=u^\flat

It is one number: the signed tendency to move around that closed curve. Different loops can have different circulations.

2D · PATCH DENSITY

Vorticity asks about local spinning.

\omega^{(2)}=d\eta\in\Omega^2(S),\qquad \zeta=\star_g\omega^{(2)}\in\Omega^0(S)

Intrinsically, vorticity is the two-form $\omega^{(2)}$ . Its Hodge dual ζ is the signed scalar used to color an oriented surface. For an embedded surface, $\boldsymbol\omega=\zeta\,\mathbf n$ . Integrating the two-form—not the normal arrow—over a patch gives its boundary circulation.

STOKES · AND ITS LIMIT

Vorticity recovers boundary circulation—not every period.

\oint_{\partial A}\eta=\int_A d\eta

On a torus, a loop around a handle is not the boundary of a patch. Its circulation is extra harmonic data that $\omega=d\eta$ cannot recover.

THE TARGET MOTION

u=-y\,\partial_x+x\,\partial_y

An arrow at each point tells a particle where to move. This velocity is already divergence-free.

THE SAME FLOW, REPACKAGED

From motion to circulation to vorticity.

Each row keeps some information convenient and makes other information less direct.

Velocity vector
$u=-y\,\partial_x+x\,\partial_y$
This is what the advection step needs.
Velocity one-form
$\eta=u^\flat=-y\,dx+x\,dy$
The one-form measures motion along a tangent direction; integrated around a loop, it measures circulation.
Vorticity two-form
$\omega^{(2)}=d\eta=2\,dx\wedge dy,\qquad \zeta=\star_g\omega^{(2)}=2$
The two-form records intrinsic local circulation density. The scalar ζ is its oriented coefficient; multiplying ζ by the chosen unit normal gives the familiar extrinsic vorticity vector. None of these tells a particle which tangent direction to move.
Clebsch variables
$\alpha=2x,\quad\beta=y,\quad\phi=-xy,\qquad d\phi+\alpha d\beta=\eta$
Three nonunique scalar fields reproduce the velocity one-form, while $d\alpha\wedge d\beta=\omega$ .

Three choices for the state

What could the solver store?

There is no universally best representation. The benefit of Clebsch variables appears only when their scalar-label structure is useful.

VELOCITY OR VELOCITY COVECTOR

Store what moves the particles.

The most direct choice is $u$ , or equivalently $\eta=u^\flat$ once a metric is fixed.

BENEFIT

No reconstruction is needed before advection.

RESPONSIBILITY

The numerical update must still preserve incompressibility and circulation.

VORTICITY TWO-FORM

Store the rotation directly.

The two-form $\omega=d\eta$ is natural when vortex transport is the central concern.

BENEFIT

Local rotation is represented explicitly.

RESPONSIBILITY

Before moving particles, reconstruct a velocity satisfying $d\eta=\omega$ and the intended divergence. On genus $g$ , also supply the $2g$ harmonic periods that vorticity forgot.

CLEBSCH POTENTIALS

Store scalar labels whose crossing carries vorticity.

\eta_C=d\phi+\alpha d\beta,\qquad d\eta_C=d\alpha\wedge d\beta

The labels can be carried with the fluid and arise naturally in some variational or Hamiltonian formulations.

BENEFIT

The curl identity is built into the representation.

RESPONSIBILITY

The potentials are nonunique, can have global topological issues, and do not enforce zero divergence by themselves.

Why projection still appears

Clebsch gives a candidate, not automatically a fluid velocity.

BUILD THE CANDIDATE
Its vorticity is already known.
The labels determine $\omega=d\alpha\wedge d\beta$ , but the raw one-form may contain a source–sink component.
$d\eta_C=\omega,\qquad \delta\eta_C\ne0\ \text{in general}$
CHOOSE THE EXACT PART
Solve one scalar Poisson problem.
Subtract an exact one-form whose divergence matches the candidate. In Clebsch language this changes only φ.
$\delta dp=\delta\eta_C,\qquad \eta=d(\phi-p)+\alpha d\beta$
USE THE RESULT
Move particles with the projected velocity.
The correction removes divergence without changing vorticity or any closed-loop circulation: $d^2p=0$ and $\oint_Cdp=0$ .
$\delta\eta=0,\qquad d\eta=\omega,\qquad \oint_C\eta=\oint_C\eta_C$

Tutorial 02

Now watch every object appear on a surface.

The second page keeps the experiments together. A third, optional reference page now holds the exterior-calculus and mesh bookkeeping.

CONTINUE TO PART IIFields in action, charts, and topology →

A fluid mustmove and remember.

Circulation asks about one chosen loop.

Vorticity asks about local spinning.

Vorticity recovers boundary circulation—not every period.

Store what moves the particles.

Store the rotation directly.

Store scalar labels whose crossing carries vorticity.

Its vorticity is already known.

Solve one scalar Poisson problem.

Move particles with the projected velocity.

Now watch every object appear on a surface.

A fluid must
move and remember.