Navier–Stokes does not fail accidentally. It is constructed to evolve local fields while explicitly discarding the information required to close its nonlinear cascade.
Σ-solutions restore that information by tracking transport across scales rather than forbidding it.
Σ-solutions restore that information by tracking transport across scales rather than forbidding it.
The Built-In Prohibition
The incompressible Navier–Stokes equations:
$$ \partial_t \mathbf{u} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u} $$
are local in space, time, and scale. The nonlinear term transports energy across scales, but the equation provides no state variable to store that transport.
Information exits the resolved scale without a bookkeeping channel to return it.
What Is Actually Missing
Navier–Stokes forbids:
- history of frame transport
- phase coherence across scales
- loop closure conditions
This is not an approximation error. It is a structural exclusion.
Σ-Solutions: Restoring Transport Accounting
A Σ-solution augments Navier–Stokes with an explicit transport sum over unresolved frames:
$$ \mathbf{u}(x,t) = \sum_{k \le k_c} \mathbf{u}_k + \Sigma_{k > k_c} \mathcal{T}_k $$
where $\mathcal{T}_k$ represents net transport across closed and open loops at scale $k$.
Key Properties
- No new forces are introduced
- No stochastic terms are required
- Transport history is preserved, not modeled
Closure Criterion
$$ \oint_k \mathcal{T} = 0 \; \Rightarrow \; \text{laminar or coherent flow} $$
$$ \oint_k \mathcal{T} \neq 0 \; \Rightarrow \; \text{cascade and turbulence} $$
Turbulence appears precisely where closure fails — not where equations break.
Why Large-Eddy Simulation Works (Partially)
LES succeeds because it approximates $\Sigma_{k > k_c}$ statistically.
It fails to be predictive because it still forbids loop history.
Transport-First Interpretation
Turbulence is saturated transport non-closure. Navier–Stokes is exact at the differential level but incomplete at the transport level.
Σ-solutions do not modify the equations — they complete them.
Σ-solutions do not modify the equations — they complete them.
Σ-Solutions vs Reynolds Averaging
Reynolds averaging replaces unresolved transport with statistical moments (e.g., Reynolds stresses), enforcing closure by assumption and sacrificing path history; it predicts mean fields but cannot reconstruct dynamics. Σ-solutions instead retain transport accounting explicitly: unresolved scales contribute through summed loop transport without invoking stochasticity or new stresses. Where Reynolds averaging collapses information into parameters, Σ-solutions preserve non-closure itself as the state. Both recover laminar limits, but only Σ-solutions remain operational across regimes because closure is tested, not imposed.
Placement on the Master Map
- Primitive: momentum transport
- Closure: fails at all scales
- Regime: saturated non-closure