One experiment · five layers
Start with a narrowing pipe. Keep the flux fixed, solve the slip-wall Euler problem, distinguish the velocity vector from its one-form, add vorticity with two scalar labels, and finally carry those labels with the flow.
The problem, then the representation
These are different jobs. Pressure enforces incompressibility in the Euler evolution; the spatial Clebsch reconstruction solves for a velocity potential φ. They are related by the constrained equations, but they are not the same scalar.
The solver begins with the closed one-form dx, Hodge-projects it against the curved walls, and keeps its period around the identified pipe. This is the two-dimensional potential-flow answer—not a resampled one-dimensional arrow field.
Pressure falls where the harmonic field accelerates and recovers when the channel widens. Because the model is inviscid, there is no permanent head loss.
At each time, the lab solves for φ so the reconstructed velocity is divergence-free and tangent to the wall. The coefficient cH is adjusted so the total harmonic period stays fixed; a single-valued φ cannot control that period.
Both du♭ and dα ∧ dβ are two-forms on this planar surface. Relative to the chosen area form they can be displayed as a signed scalar density; as an ambient picture, the associated vorticity vector points normal to the plane.
Vector or covector?
Feed the tangent vector u to the advection ODE. In Chapter 3 it is drawn as a cyan arrow.
Feed the one-form u♭ a test displacement. The orange bar is its kernel: moving along that bar contributes zero.
The Euclidean metric makes their components look identical here, but the objects still transform differently and do different jobs.
Chapter 5 · the algorithm
The simulation repeats a material transport step followed by a spatial Hodge reconstruction. This makes the division of labor visible.
Next: shallow water on a surface
Here the wall shape A(x) is prescribed and divergence is projected away. In shallow water, the layer depth h evolves, horizontal convergence changes h, and pressure comes from the hydrostatic potential one half g h squared.