Same local curl · different global transport

What did
vorticity forget?

On a periodic square, adding a constant flow changes how particles wind around the torus but changes no local vorticity. The missing information is exactly two numbers: circulation around the horizontal cycle and circulation around the vertical cycle.

PATCH DATA

Vorticity fixes local boundary circulation.

\omega=d\eta,\qquad \oint_{\partial A}\eta=\int_A\omega

For every contractible loop that bounds a patch A, vorticity determines the circulation.

LOOP DATA

Two noncontractible periods remain.

\Gamma_x=\oint_{\gamma_x}\eta=c_x,\qquad \Gamma_y=\oint_{\gamma_y}\eta=c_y

Neither generator is a patch boundary. These two numbers are independent velocity data.

HARMONIC DATA

The periods select one velocity among many.

\eta=d^+\omega+c_x\,dx+c_y\,dy

All choices share the same vorticity. Pressure cannot choose between them because subtracting $dp$ changes no closed-loop period.

Read the experiment

01
Start from vorticity
The intrinsic object is the Taylor–Green two-form $\omega^{(2)}=d\eta$ . The background color shows its Hodge-dual scalar $\zeta=\star_g\omega^{(2)}$ ; together they determine the coexact velocity.
02
Add two missing numbers
The periods $c_x,c_y$ choose the constant harmonic flow around the two torus cycles.
03
Compare c = 0 with c ≠ 0
Both fields have identical vorticity. The nonzero harmonic state adds uniform transport around the periodic cycles.
04
Only then try quantization
The collapsed optional panel introduces qℤ². Its “nearest” point is a rounding rule after choosing q, a basis, and a norm—not a fluid law.

DO NOT CALL THIS A GAUGE TRANSFORMATION Subtracting a harmonic field changes the fluid.

It preserves curl and divergence, but changes flux through a cross-section and particle winding around the torus.

PERIODIC FUNDAMENTAL DOMAINopposite edges are identified

raw material after subtraction

ARROWS

MATERIAL GRID

SELECTED RECONSTRUCTION

(c_x,c_y)-q(0,0)=(0.85,-0.55)

No lattice field is removed. Both grids coincide and carry the same nonzero physical harmonic drift.

raw periods—

removed field—

residual periods—

residual harmonic energy—

raw mean winding—

cyan mean winding—

The real harmonic state

Horizontal period cₓcirculation around the horizontal generator0.85 Vertical period cᵧcirculation around the vertical generator−0.55 Vortex strength Achanges local curl, not either period0.72

START WITH THE PHYSICAL STATE

The evolving-label demo has c = (0,0): local Taylor–Green deformation with no uniform drift. The nonzero example changes only the two global periods.

TWO GENERATORS OF H¹(T²)

dxunit horizontal period

dyunit vertical period

A harmonic field on this flat unit torus is constant. Its two coefficients are real-valued physical state; neither vorticity nor pressure determines them.

OPTIONAL EXTENSIONQuantized harmonic subtractionOpen only after cₓ and cᵧ make sense.

Now impose an additional lattice $q\mathbb Z^2$ and compare the fluid with another one whose harmonic state differs by a lattice vector:

c'=(c_x,c_y)-q(m,n),\qquad(m,n)\in\mathbb Z^2

This subtraction changes the physical fluid. It is not a Clebsch gauge choice, pressure projection, or required step of fluid cohomology.

Circulation quantum qan imposed spacing, not determined by the fluid0.50

WHAT “NEAREST” MEANS HERE

m_* = \operatorname{round}(c_x/q),\qquad n_* = \operatorname{round}(c_y/q)

This minimizes $\lVert c-q(m,n)\rVert^2$ only because dx and dy form the chosen orthogonal unit basis. Change q, the harmonic basis, or its metric Gram matrix, and “nearest” can change. There is no canonical nearest point before making those choices.

INTEGER FIELD TO SUBTRACT(m,n) = (0,0)

Two different zeros: (m,n) = (0,0) means “remove nothing.” It does not set the physical state c to zero.

What the subtraction means

Three operations that look similar—but are not.

All three manipulate one-forms. Only one is a pressure projection, and only one changes the cohomology class.

01 · PRESSURE

Remove an exact field

\eta\mapsto\eta-dp

Enforces zero divergence. Closed-loop periods and vorticity are unchanged because an exact form integrates to zero around every closed loop.

02 · STREAMFUNCTION

Recover the coexact field

\omega\mapsto d^+\omega

Recovers the least-norm velocity seen by vorticity. “Least norm” silently sets both harmonic periods to zero; it does not infer them.

03 · OPTIONAL QUANTIZATION

Change the harmonic state

(c_x,c_y)\mapsto(c_x-qm,c_y-qn)

Preserves divergence and vorticity, but changes cross-sectional flux. “Nearest” means minimum residual energy only after this lab imposes q, the basis dx,dy, and the flat metric.

Primary source: Hang Yin, Mohammad Sina Nabizadeh, Baichuan Wu, Stephanie Wang, and Albert Chern, Fluid Cohomology, ACM Transactions on Graphics 42(4), 2023. The integer lattice in this tutorial is an added mathematical experiment, not a claim made by the paper.