3D Galerkin NS with Vortex Stretching
Overview
The critical difference between 2D and 3D Navier-Stokes: in 3D, the vortex stretching term \((\omega \cdot \nabla) u\) can amplify vorticity without bound. This experiment uses a 6-mode Galerkin truncation of the 3D vorticity equation — 3 velocity modes at wavenumber \(k_1 = 1\) and 3 vorticity modes at \(k_2 = 2\) — with stretching coupling \(\lambda\) and feedback coupling \(\sigma\).
Key Results
Model Parameters
| Parameter | Role | Effect |
|---|---|---|
| \(\sigma\) | Vorticity → velocity feedback | Drives velocity via \(\omega \times u\) |
| \(\lambda\) | Linear vortex stretching | \(\lambda = 0\): 2D-like; \(\lambda > 0\): 3D stretching |
| \(\lambda_2\) | Quadratic self-amplification | \(\lambda_2 |\omega|^2 \omega\): nonlinear enstrophy cascade |
| \(\nu\) | Viscosity | Diffusive stabilisation at \(k^2\) scaling |
Diagnostic Survival in 3D
| Diagnostic | 2D Result | 3D Result | Survives? |
|---|---|---|---|
| \(|\partial T / \partial A| \to \infty\) as \(A \to 0\) | Yes | Yes | Yes |
| I-ratio \(= -0.5\) at equilibrium | Yes | Yes (diffusion vs stretching) | Yes |
| \(S\)-score early warning | 37× lead | Reduced but present | Yes |
| Stability margin \(M(t)\) | Positive | Positive (reduced by stretching) | Yes |
Interpretation. All four diagnostic tools developed on 1D and 2D models survive the transition to 3D with vortex stretching. The regularity signature \(|\partial T / \partial A| \to \infty\) is the strongest — it holds even at moderate stretching strengths. The stability margin is reduced by stretching but remains positive, indicating viscosity still dominates at these parameters.
Analysis
- 6-mode model captures the essential 3D mechanism. The vortex stretching term \((\omega \cdot \nabla)u\) is faithfully represented in the Galerkin truncation. The sign-indefinite nature of stretching (it can amplify or diminish vorticity depending on alignment) is preserved.
- I-ratio generalises to 3D. In 3D, the I-ratio measures diffusion vs stretching rather than diffusion vs advection. The equilibrium condition \(I = -0.5\) still holds.
- Stretching reduces but does not eliminate the stability margin. Viscosity at \(k_2^2 = 4\) scaling on the vorticity modes provides sufficient damping at the parameters tested. The question is what happens at extreme \(\lambda\) — see the breaking-point experiment.
Reproducibility
../simplex/build/sxc exp_ns_3d_vortex.sx -o build/exp_ns_3d_vortex.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_3d_vortex.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_3d_vortex \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_3d_vortexRelated
- Dual Agent Architecture — order vs chaos on 1D Burgers
- Where Does Regularity Break? — pushing stretching to extremes
- 2D Four-Mode NS — the 2D baseline