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ADVANCED UNDERGRADUATE PATHClebsch fluids on surfaces

Ten lessons · four conceptual turns

Learn the language.
Find the topology.
Then evolve the fluid.

This path is for students building toward shallow water on curved surfaces with Clebsch variables. Each lesson asks one question, introduces one new burden, and preserves the audits established earlier.

Begin lesson 1See the complete path
ASSUMEDVector calculus, linear algebra, and ODEs

No prior differential geometry is required. The first unit builds the necessary exterior-calculus language from measurements.

NOT ASSUMEDThat Clebsch variables are coordinates

They are nonunique scalar fields used to factor a velocity one-form. They become local coordinates only where their differentials are independent.

ENDPOINTA solver architecture, not one magic formula

Students should know which state is transported, which elliptic problem is solved, and which harmonic data topology leaves undetermined.

NOTATION GUARDRAIL
φ

Clebsch velocity potential. It is not the pressure.

p

Euler pressure or a constraint multiplier.

ηk

Harmonic basis one-forms carrying global periods.

h

Water-column depth in the shallow-water unit.

UNIT I · READ THE FIELD

From arrows to circulation.

Do not begin with three mysterious potentials. First learn what a velocity vector does, what its metric-dual one-form measures, and what exterior differentiation extracts.

vector to covector to vorticity
  1. 01
    FORMS, d, AND THE HODGE STAR

    What kind of object lives where?

    Distinguish vertex scalars, edge-integrated one-forms, face-integrated two-forms, and metric-dependent conversions.

    Exit test: identify the type and integration domain of every variable.
    OPEN LAB →
  2. 02
    THE PHYSICAL PROBLEM

    What must a fluid representation remember?

    Separate velocity, circulation, vorticity, and incompressibility before comparing velocity, vorticity, and Clebsch states.

    Exit test: explain why circulation is not the same datum as pointwise vorticity.
    OPEN LESSON →
  3. 03
    CLEBSCH IN ACTION

    How do α, β, and φ share the work?

    Carry material labels, make the covector visible, assemble the one-form, and inspect the first chart obstruction.

    Exit test: derive du♭ = dα ∧ dβ and state what dφ cannot change.
    OPEN LAB →
UNIT II · FIND WHAT LOCAL VORTICITY MISSES

Boundaries, holes, and periods.

Stokes determines circulation around patch boundaries. It does not determine circulation around loops that bound no patch. That distinction becomes solver state.

Hodge decomposition with topology
  1. 04
    DISK → ANNULUS

    When does circulation have to be vorticity?

    Use a movable Stokes loop, then remove the center and separate vortical circulation from one harmonic degree of freedom.

    Exit test: predict which combination of inner and outer circulation changes curl.
    OPEN LAB →
  2. 05
    FLAT TORUS

    Which two numbers does vorticity forget?

    Hold local vorticity fixed while changing the two real harmonic periods and the resulting particle winding.

    Exit test: explain why pressure projection cannot erase a harmonic period.
    OPEN LAB →
UNIT III · SEPARATE KINEMATICS FROM DYNAMICS

Prescribe, project, then transport.

A beautiful moving field is not automatically a fluid solution. First inspect random kinematic fields; then let Euler transport the representation that generates vorticity.

material labels plus constraint
  1. 06
    RANDOM SURFACE FIELDS

    What can we learn without claiming dynamics?

    Combine intrinsic Laplace–Beltrami modes with temporal Perlin coefficients; compare exact, coexact, raw Clebsch, and projected Clebsch families.

    Exit test: distinguish a prescribed time-dependent velocity from a field evolved by Euler.
    OPEN LAB →
  2. 07
    BERNOULLI → CLEBSCH TRANSPORT

    How do pressure, topology, and labels coexist?

    Start with periodic potential flow through a narrowing channel, recover Bernoulli pressure, add label vorticity, and re-solve φ after transport.

    Exit test: write one time step as transport, candidate construction, Hodge solve, and period restoration.
    OPEN LAB →
UNIT IV · LET DEPTH AND GEOMETRY EVOLVE THE STORY

From incompressible flow to shallow water.

Do not project horizontal divergence to zero once it is responsible for changing layer depth. Preserve the representation lessons while replacing the constraint.

depth-weighted continuity
  1. 08
    CONSERVATIVE BASELINE

    Which audits survive a change of variables?

    Test a linear pressure-wave model, its gradient/divergence adjoint pair, mass, energy, CFL behavior, and a deliberately separate passive dye.

    Exit test: explain why shallow-water divergence changes h instead of being projected away.
    OPEN LAB →
  2. 09
    CLEBSCH SHALLOW WATER

    How do label transport and height flux couple?

    Follow the scalar → covector → vector → flux pipeline and keep the Clebsch and continuity equations conceptually distinct.

    Exit test: identify which variables are material, which are diagnostic, and which elliptic solve remains.
    OPEN LAB →
  3. 10
    NON-ORIENTABLE EXTENSION

    What changes when no global normal exists?

    Carry water through a reflected seam, distinguish twisted vorticity from ordinary scalars, use an O(2) frame, and audit material Kelvin loops.

    Exit test: separate local connection terms from global orientation holonomy.
    OPEN LAB →

References and branches

Useful, but not prerequisites for the spine.

REFERENCEExterior calculus + DEC lookup

Return when a type, discrete storage location, sign convention, or gauge becomes ambiguous.

 ELECTIVELine and polyvector Clebsch fields

Study unordered branches and monodromy after ordinary signed velocity fields are secure.

 IMPLEMENTATIONEditable sparse energies

Explore the optimization and differentiation infrastructure used elsewhere in the lab.

 WORKFLOWRun, fork, and publish

Use this when moving from a browser lesson to a student code project.