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Theory

The Commutativity Problem on the Sphere

Aluie (2019, doi:10.1007/s13137-019-0123-9) proves that on the sphere S², filtering vector fields by converting to Cartesian components and filtering each as a scalar does NOT commute with differential operators:

G * (∇·u) ≠ ∇·(G * u)    on S²
G * (∇×u) ≠ ∇×(G * u)    on S²

This means that filtering a divergence-free velocity field via Cartesian components does NOT produce a divergence-free filtered field.

The Helmholtz Solution

Aluie (2019, Section 7, Proposition 2) proves that the generalized convolution that DOES commute with derivatives is mathematically equivalent to:

  1. Decomposing u into scalar potentials: u → (ψ, χ, u_r)
  2. Filtering each scalar separately: ψ̄, χ̄, ū_r
  3. Reconstructing velocity from filtered potentials

This is the Edmonds transformation (Edmonds 1960).

The full Helmholtz–Hodge decomposition

On a domain Ω ⊂ ℝⁿ, a vector field admits the L²-orthogonal decomposition

u  =  ∇χ  ⊕  rot R  ⊕  h

into a curl-free part ∇χ (divergent), a divergence-free part rot R (rotational), and a harmonic part h that is both divergence- and curl-free. On a periodic torus or an unbounded domain with decay, dim of the harmonic space is 0 (only the mean, which the solvers remove). On a bounded or multiply-connected domain the harmonic space has dimension equal to the first Betti number b₁ of the domain (the number of holes/islands): h then carries the net circulation around each hole and the net flux through it, which no single-valued ψ/χ can represent.

The decomposition is not unique on a bounded domain without boundary conditions. The "natural HHD" (Bhatia et al. 2013) fixes uniqueness and orthogonality by assigning the normal trace to ∇χ (Neumann ∂χ/∂n = u·n̂) and the tangential trace to rot R (Dirichlet ψ = const per boundary component). This package computes h as the residual h = u − ∇χ − rot R after the potential solves (harmonic_fraction = ‖h‖/‖u‖ is reported), and detects topology with count_holes / betti1_estimate.

N-dimensional potentials

The 3-D vector potential generalizes to an antisymmetric rotation-potential matrix R with N(N−1)/2 independent components (Glötzl & Richters 2023). The governing equations are component-wise Poisson problems,

Δχ      = ∇·u                              (gradient/scalar potential)
ΔR_ab   = ∂_a u_b − ∂_b u_a   (a < b)      (rotation potential)

with the identity grad div u + ROT ROT̄ u = Δu generalizing grad div − curl curl = Δ. Per dimension: 1-D → pure gradient (no rotation); 2-D → one component R₁₂ = ψ (streamfunction); 3-D → three components, the Hodge dual of the vector potential A (u_rot = ∇×A); N ≥ 4 → only χ and the matrix R are well-defined (no ψ, no A). In Fourier space the velocity split is dimension-trivial — the Leray projection û_div = (k̂⊗k̂)û, û_rot = (I − k̂⊗k̂)û — which is the fast path used by the spectral solvers.

The 2D special case

Any 2D vector field can be written u = u_rot + u_div (+ harmonic), with u_rot = ∇×(ψ ẑ) non-divergent and u_div = ∇χ irrotational.

Cartesian Geometry

Given vorticity ζ = ∂v/∂x − ∂u/∂y and divergence δ = ∂u/∂x + ∂v/∂y:

  • Solve ∇²ψ = ζ
  • Solve ∇²χ = δ
  • urot = −∂ψ/∂y, vrot = ∂ψ/∂x
  • udiv = ∂χ/∂x, vdiv = ∂χ/∂y

Spherical Geometry

On a sphere of radius R with coordinates (λ, φ):

  • Solve ∇²ψ = ζ where ∇² is the spherical Laplacian
  • u_rot = −(1/R) ∂ψ/∂φ
  • v_rot = 1/(R cos φ) ∂ψ/∂λ

The spherical Laplacian has eigenvalues −ℓ(ℓ+1)/R² for spherical harmonic degree ℓ.

Spectral Poisson Solvers

The Poisson equation can be solved in O(N log N) time using spectral methods:

  1. Transform RHS to spectral space (FFT or SHT)
  2. Divide each mode by its eigenvalue (−k² for FFT, −ℓ(ℓ+1)/R² for SHT)
  3. Transform back to physical space

The k=0 (or ℓ=0) mode is set to zero (the mean of the solution is arbitrary).

References

  • Aluie, H. (2019). Convolutions on the sphere: commutation with differential operators. GEM - International Journal on Geomathematics, 10(1), 9. doi:10.1007/s13137-019-0123-9
  • Glötzl, E., & Richters, O. (2023). Helmholtz decomposition and potential functions for n-dimensional analytic vector fields. Journal of Mathematical Analysis and Applications, 525(2), 127138. doi:10.1016/j.jmaa.2023.127138
  • Bhatia, H., Norgard, G., Pascucci, V., & Bremer, P.-T. (2013). The Helmholtz-Hodge Decomposition — A Survey. IEEE TVCG, 19(8), 1386–1404. doi:10.1109/TVCG.2012.316
  • Buzzicotti, M., Storer, B. A., Khatri, H., Griffies, S. M., & Aluie, H. (2023). Spatio-temporal coarse-graining decomposition of the global ocean geostrophic kinetic energy. Science Advances, 9(45). doi:10.1126/sciadv.adi7420
  • Storer, B. A., Buzzicotti, M., Khatri, H., Griffies, S. M., & Aluie, H. (2022). Global energy spectrum of the general oceanic circulation. Nature Communications, 13, 5314. doi:10.1038/s41467-022-33031-3