Stack intrinsic scales. Modulate them with smooth time noise. Advect.

Random fluids,
bound to a surface.

Build one seeded scalar field from intrinsic surface eigenmodes, let its coefficients wander smoothly in time, then choose its exact gradient or coexact quarter-turned gradient. This is a kinematic playground: it isolates representation and projection before a physical evolution law is added. The Clebsch construction has its own focused tutorial.

Open the live surface demo ↓

THE CONSTRUCTION

f(x,t)=\sum_j c_j(t)\,\varphi_j(x),\qquad -\Delta_S\varphi_j=\lambda_j\varphi_j

u=J\nabla_S\psi

tangent

u\cdot n=0

area preserving

\operatorname{div}_S u=0

Every intrinsic scalar mode is rotated by a quarter-turn in its tangent plane. The temporal Perlin coefficients change amplitudes, but do not break the coexact field's divergence-free identity.

Guided derivation · about five minutes

From intrinsic modes to a random surface flow.

The geometry and randomness have separate jobs. Laplace–Beltrami eigenfunctions supply spatial patterns intrinsic to the surface. Smooth one-dimensional noise changes their coefficients in time. A Hodge choice then turns the scalar into either exact or coexact motion. None of these choices is yet a shallow-water time step.

01TIME COHERENCE

Perlin noise moves coefficients—not geometry.

The spatial functions are fixed first. The sphere uses real spherical harmonics. The tree frog uses eigenvectors of its cotangent Laplace–Beltrami operator. The periodic square uses Fourier eigenfunctions; the parameterized torus uses periodic coordinate modes. No ambient 3D plane wave is restricted onto either curved mesh.

f(x,t)=\sum_j A_j\bigl(0.62+0.78N(o_j+r_jt)\bigr)\,\varphi_j(x)

fixed intrinsic mode + smooth temporal Perlin coefficient

K\varphi_j=\lambda_jM\varphi_j\quad\text{(triangle mesh)}

On the frog,

K

and

M

come from the actual triangles, so the modes depend on intrinsic mesh geometry—not its pose in space.

1Hash a random slope at each integer time lattice point.

2Take each slope’s linear contribution inside the current interval.

3Blend them with

6\xi^5-15\xi^4+10\xi^3

for smooth joins.

Each eigenfunction follows an independent 1D noise path $N(o_j+r_jt)$ . The turnover slider scales $r_j$ ; finer bands also evolve faster by roughly $\sqrt{\mathrm{band}}$ . The eigenvalue $\lambda_j$ supplies the spatial frequency $k_j=\sqrt{\lambda_j}$ . At zero turnover, the random field freezes exactly.

02HODGE FAMILY

One scalar gives two very different flows.

Taking the surface gradient produces the exact family. It crosses contours, has zero curl, and can focus or disperse particle clouds.

u_{\parallel}=\nabla_S f,\qquad \operatorname{curl}_S u_{\parallel}=0

exact does not mean incompressible: its divergence is the surface Laplacian of the scalar.

Rotate the gradient by a quarter-turn in the oriented tangent plane and it becomes coexact:

u_{\perp}=J\nabla_S f,\qquad \operatorname{div}_S u_{\perp}=0

The velocity now follows scalar contours instead of crossing them.

The randomness is identical in both cases. Switching the field construction isolates the geometric effect of the quarter-turn $J$ .

Topology audit: scalar gradients and rotated scalar gradients carry no independently prescribed harmonic periods. On the torus this lab therefore explores the zero-harmonic sector. The flat-torus lab adds the two loop circulations that vorticity cannot see.

03SCALE SPECTRUM

Eigenvalue means spatial frequency.

Laplace–Beltrami modes order patterns from broad to fine. The lab converts each eigenvalue into a wave number and weights it with the selected energy law.

geometry $-\Delta_S\varphi_j=\lambda_j\varphi_j$
frequency $k_j=\sqrt{\lambda_j}$
energy $E(k)\propto k^{-\beta}$

LOW BANDSbroad coherent structures

HIGH BANDSsmall rapidly changing eddies

SLOPE βbalance of coarse and fine energy

On the frog, the generalized eigenproblem uses cotangent stiffness and lumped vertex area. This makes the random field depend on the triangle metric rather than on an arbitrary ambient restriction.

04SAMPLE + ADVECT

Then ask what the flow does to material.

The chosen exact or coexact field is reconstructed smoothly over the surface. Particles query that velocity at their own positions and advance with a midpoint step, wrapping periodic coordinates or renormalizing to the sphere.

\dot{x}(t)=u(x(t),t)

the two particle colors are passive material labels, not different dynamics.

The arrows are a separate observation layer. Readable vertices skips some actual mesh vertices; Every vertex evaluates the velocity at every unique rendered vertex. Neither mode invents a second display grid.

divergence RMS ↓ less local area change vorticity RMS strength of local rotation correlation ↓ field has decorrelated from time zero

How to read the experiment

01
Hold the realization fixed.
A seed fixes the spatial modes, temporal noise paths, and initial particle clouds, so comparisons are repeatable.
02
Compare exact with coexact.
Use the same seed. The gradient crosses contours and can compress; its quarter-turn follows contours and preserves area.
03
Then animate the coefficients.
Turnover moves the modal weights along smooth noise paths. Field correlation measures memory of the initial velocity.

SURFACE

HODGE FAMILY

Where does the random field live?

Its spatial basis lives intrinsically on the selected surface. Only the modal coefficients are random in time. The mesh, eigenfunctions, and tangent planes remain fixed.

Why can the exact flow collapse?

A gradient is curl-free, not divergence-free. Its surface divergence is $\Delta_S\phi$ , so it can create local sources and sinks for the particle density.

Where did Clebsch go?

It has a dedicated three-part Clebsch variables on surfaces tutorial: first the motivating fluid problem, then the live construction and torus obstruction, followed by a compact exterior-calculus and DEC reference. This page now isolates spectral randomness and the exact/coexact distinction.

LIVE TANGENT FIELD · MESH VERTICEStree frog · coexact / divergence-free

drag to orbit · scroll to zoom

cloud A cloud B

time0

RMS speed—

divergence RMS—

vorticity RMS—

tangent defect—

field correlation—

Seed 13 · coexact spectral field ready on tree frog

Random ensemble

VECTOR GLYPHS · ACTUAL MESH VERTICES Choose mesh-vertex coverage

Both modes evaluate the analytic velocity directly at rendered mesh vertices. Readable mode only skips glyphs; it does not resample or interpolate the field.

Mesh vertex counts update with the selected surface.

Seedreplay this exact realization Energy falloff β

E(k)\propto k^{-\beta}

1.67 Active modesindependent seeded coefficients 28 Highest bandsmallest visible eddy scale 7 Time-Perlin ratezero freezes every noise coordinate 0.42 RMS speedcalibrated over the whole surface 0.62 Particlessplit between two material clouds 1000

INPUT SCALE SPECTRUMβ = 1.67 · 7 bands

CHECK 01 · GEOMETRY

The modes belong to the surface.

On the frog, they are eigenvectors of the cotangent Laplace–Beltrami operator. On the sphere and periodic parameter surfaces, they come from the corresponding intrinsic harmonic families.

CHECK 02 · HODGE

A quarter-turn changes the local invariant.

The exact gradient is curl-free and generally compressible. Its coexact quarter-turn is divergence-free and vortical. On a torus, neither construction supplies the two independent harmonic circulations.

CHECK 03 · PERLIN

Smooth randomness is not fluid dynamics.

Temporal Perlin noise prescribes how the random field changes; it is not a Navier–Stokes or shallow-water evolution. The particles respond to that kinematic velocity, but they do not feed back into it.

Random fluids,bound to a surface.