A periodic plane · material labels · one-forms · shallow water

Clebsch variables,
without the sleight of hand.

The variables are scalar fields. They do not become a velocity until their differentials are assembled into a one-form and the metric raises that covector to a vector. This lab keeps every step visible on one planar grid.

Open the live grid ↓

THE REPRESENTATION

u^\flat=d\phi+\alpha\,d\beta

u=(u^\flat)^\sharp

On this Cartesian plane, raising an index preserves the two displayed components. That numerical coincidence is why vectors and covectors are easy to conflate here.

HEIGHT / DENSITYh(x,y,t)

Water column carried by the flux hu. Pressure comes from its gradient.

BERNOULLI POTENTIALφ(x,y,t)

Its differential dφ supplies the exact, curl-free part of the velocity one-form.

LABEL WEIGHTα(x,y,t)

A materially transported scalar that weights how strongly the β-label contributes.

MATERIAL LABELβ(x,y,t)

Another transported scalar. Its level curves move with the water.

THE IMPORTANT CHANGE OF PROBLEM

Do not project shallow water to zero divergence.

In the random-flow lab, subtracting $dp$ is a useful incompressible comparison. In shallow water, convergence and divergence are physical: they raise and lower the water column. Pressure acts through the evolution of φ while the continuity equation evolves h.

\partial_t h+\operatorname{div}(hu)=0

D_t\phi=\tfrac12|u|^2-g(h-H)

On a torus, the velocity state may also carry two harmonic periods. This planar lab sets them to zero; a topology-aware solver must retain and audit them separately.

How to read the grid

01
Begin with β.
The colors are scalar values. Neighboring colors tell you the covector $d\beta$ .
02
Switch to covector bars.
Each little stack is tangent to the kernel of the one-form. Crossing the bars measures change.
03
Raise the index.
Velocity arrows are $(d\phi+\alpha d\beta)^\sharp$ .
04
Follow the labels.
Play the flow: α and β deform but remain material scalars.

So which things are vectors?

u and the mass flux hu are vectors. The gradients drawn as arrows are metric-dual vectors used to visualize the covectors dφ and dβ. The primitive Clebsch variables themselves are scalars.

Why draw a covector as bars?

A covector assigns a number to every test displacement. Its kernel is the line of displacements on which that number is zero. Parallel bars make that kernel visible; their normal points in the measuring direction.

What is omitted?

This periodic example fixes both harmonic circulations to zero. Vorticity cannot reconstruct those two torus periods; a complete torus solver must store or evolve them. On a curved surface, the metric-dependent sharp map is no longer visually invisible.

LIVE PERIODIC GRIDheight h with velocity vectors

ready

GLYPHS

Arrows are vectors: they point where a particle moves.

time0

mass drift—

energy—

divergence RMS—

vorticity RMS—

ζ − dα∧dβ RMS—

THE FOUR STATE FIELDS AT ONCEClick a tile to bring it to the main grid.

The careful version

Scalar → covector → vector → transport.

Clebsch notation is compact because it suppresses those type conversions. Expand them, and the construction becomes much less mysterious.

0-FORMS · SCALARS

φ, α, β, h

Each assigns one number to every grid point. Their colors have no direction.

φ(x,y), α(x,y), β(x,y), h(x,y)

take d

1-FORMS · COVECTORS

dφ + αdβ

A one-form measures displacements. Its kernel bars show which displacements produce zero reading.

u♭(δx) ∈ ℝ

raise ♯

VECTORS

u

The metric turns the measuring one-form into the arrow that transports water and labels.

ẋ = u(x,t)

WHERE DOES ROTATION COME FROM?

The crossing of two scalar foliations.

d u^\flat=d(d\phi)+d(\alpha d\beta)=d\alpha\wedge d\beta

The exact piece disappears because $d^2\phi=0$ . Rotation is large where α and β change strongly in independent directions. If their contours are parallel, their wedge product vanishes.

VECTOR VS. COVECTOR

Same two numbers here; different jobs.

vector u

Consumes time and produces displacement.

covector u♭

Consumes displacement and produces a number.

In Cartesian coordinates with the identity metric, $u_i=\delta_{ij}u^j$ copies the components. On a curved or stretched mesh, the metric changes them.

THE SHALLOW-WATER SYSTEM IN CLEBSCH VARIABLES

The labels move; height compresses; φ pays the Bernoulli bill.

$D_t=\partial_t+u\cdot\nabla$ . The lab semi-Lagrangian-advects the labels and φ, advances h with conservative periodic face fluxes, and fixes the irrelevant spatial mean of φ. A weak periodic de-aliasing diffusion $\kappa=5\times10^{-4}$ suppresses grid-scale folds once the transported labels become finer than the mesh; the vorticity audit makes the remaining representation error visible.

$D_t\alpha=0,\qquad D_t\beta=0$
The two labels are painted onto the water.
$D_t\phi=\tfrac12|u|^2-g(h-H)$
Kinetic and pressure terms update the exact potential.
$\partial_t h+\nabla\cdot(hu)=0$
Height is a density, so it changes under compression.
$u=(d\phi+\alpha d\beta)^\sharp$
The one-form is rebuilt from the four scalar fields.

Clebsch variables,without the sleight of hand.