Turbulence · Cascading Transport Non-Closure

Claim: Turbulence is not randomness in fluids, but the continuous N-body limit of transport non-closure.

What Turbulence Is (Operationally)

No closed-form solution exists for turbulent flow, even with deterministic equations.

A fluid is a continuum of interacting transport frames.

Each fluid element both:

This makes turbulence the limit of infinite, coupled transport loops.

Primitive: Motion relative to evolving local frames

Consider a closed loop embedded in a flow.

Transport around the loop depends on smaller-scale vortices within it.

$$ \oint (\mathbf{v} - \mathbf{u}_{\ell}) \cdot d\mathbf{r} \neq 0 $$

As scale $\ell$ decreases:

This produces the turbulent cascade.

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Primitive: Phase accumulation under transport

Each vortex carries a phase $\theta(\ell)$.

In laminar flow, phases align.

In turbulence:

$$ \theta(\ell_1) \not\rightarrow \theta(\ell_2) $$

Phases decorrelate across scales.

No global phase ordering exists.

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$$ \text{Cascade of non-closure} \iff \text{Phase decoherence across scales} $$

Energy flux becomes the only surviving invariant.

$$ E(k) \sim \varepsilon^{2/3} k^{-5/3} $$

This is not a force law, but a transport-statistical residue.

Viscosity limits the smallest transport loop.

It halts the cascade by enforcing local closure.