D03 — Kolmogorov Turbulence Cascade
| Field | Value |
|---|---|
| Domain | Dynamical Systems |
| System | 3D incompressible Navier-Stokes, inertial range |
| Group | SO(3) × ℝ₊ (rotation + scale) |
| H^k tier | H¹ / H² |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · SPLIT · SPLAT · LABEL |
| Paper | Paper 512 |
Physical system
3D incompressible Navier-Stokes at high Reynolds number Re = UL/ν. Energy injected at the integral scale L cascades through the inertial range to the Kolmogorov dissipation scale η = (ν³/ε)^{1/4}, where ε is the energy dissipation rate. The Richardson-Kolmogorov picture: big whorls have little whorls that feed on their velocity, and little whorls have lesser whorls, and so on to viscosity (Richardson 1922).
Target category
Symp_∞ — the category of infinite-dimensional symplectic manifolds, with objects = divergence-free vector fields on ℝ³ and morphisms = symplectic maps (the Euler/NS flow). Objects are stratified by Fourier wavenumber k: the object at scale k is the symplectic leaf T*L²_k (kinetic energy at wavenumber k).
Interpretation functor
F: C → Symp_∞ defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Kolmogorov energy cascade: energy flows k → 2k → … → k_η; winding number = Re^{3/4} cascade depth |
| TWIST | Vortex stretching: ω → ω + (ω·∇)u δt; local H¹ phase accumulation in vorticity field |
| SPLIT | Turbulent energy injection: large eddy breaks into two smaller eddies |
| SPLAT | Viscous dissipation at Kolmogorov scale: two sub-eddies merge and dissipate |
| LABEL | Pressure field p: Leray projector enforcing ∇·u = 0; eigenvalue = −∇²p = tr(∂ᵢuⱼ)² |
ISA programme
INJECT: SPLIT[u_L → u_{L/2} ⊕ u_{L/2}] -- energy injection at integral scale
CASCADE: ORBIT[k → 2k | k < k_η] -- inertial range cascade (H¹ ORBIT)
STRETCH: TWIST[ω_k | vortex stretching] -- intermittency events
DISSIP: SPLAT[u_η → heat] -- viscous dissipation at η
ENFORCE: LABEL[p | ∇·u = 0] -- pressure enforces incompressibility
Kolmogorov β-value: the cascade runs at β = β_Kolmogorov = 1/ε^{1/3} L^{2/3} (the reciprocal of the inertial-range energy per unit wavenumber). The −5/3 spectrum is the fixed point of ORBIT at this β.
Computable output
- Energy spectrum: E(k) ∝ ε^{2/3} k^{−5/3} — the Kolmogorov −5/3 law. This is the eigenvalue of the ORBIT fixed point: the unique scale-invariant distribution of energy across wavenumbers. Validated by every turbulence experiment since Grant, Stewart & Moilliet (1962).
- Kolmogorov scale: η = (ν³/ε)^{1/4} — the ORBIT termination scale where SPLAT overtakes SPLIT.
- H² blow-up criterion (Tao 2016 / Paper 512): Navier-Stokes develops a finite-time singularity if and only if the rate of H² vorticity obstruction class generation (TWIST events) exceeds the rate of viscous H² class resolution (SPLAT) without bound. This reformulates the Beale-Kato-Majda criterion (∫₀ᵀ ‖ω‖_∞ dt → ∞ ⟹ blow-up) in topological language.
- Undecidability: Tao (2016) showed that for Turing-complete fluid initial conditions, the question of blow-up is undecidable — equivalent to the halting problem. In ISA terms: there is no local polynomial criterion that decides whether the ORBIT closes.
Validation
- Kolmogorov −5/3 spectrum: confirmed experimentally in ocean, atmosphere, wind tunnel, and pipe flow over 6 decades of wavenumber.
- Vortex reconnection as ±1 H² linking number change: confirmed in superfluid helium experiments (Bewley et al. 2008) and direct numerical simulation.
- Tao’s Turing completeness construction: rigorous (averaged NS), published in J. Amer. Math. Soc. (2016).