What sets the magnitude of the annular Reynolds stress relative to the collapse requirement?

Case: the unforced swirl annulus that builds the collapse geometry and stops short.

Axisymmetric NS with swirl, z-periodic, unforced, ν = 2×10⁻³, Rayleigh-unstable exterior. Over 14 time units: fluctuation energy grows e¹³ and saturates; annular Reynolds stress ⟨u_r′u_θ′⟩ positive (+0.64) — outward angular-momentum flux; its divergence exceeds viscous diffusion (0.044 vs 0.038); saturated state is a single meridional cell; the local swirl maximum decouples from the z-mean, sits 7.8% above it, and stops decaying; its radius does not move, r ≈ 0.5.

Friction. Account A — a Rayleigh-unstable profile mixes itself flat, nothing collapses. But the instability produces the collapse geometry with the right stress sign, and the core stops decaying. Account B — the instability sustains the collapse, as in the forced construction. But the cell delivers inflow strain ≈ 1, fixed by cell size, where collapse requires 4/τ growing without bound. The stress direction is pinned by energetics; the magnitude is unexplained.

Question for the panel. What sets the magnitude of the annular Reynolds stress relative to the collapse requirement in an unforced flow — Reynolds number alone (a threshold Re_θ above which cell strain outruns core diffusion), non-axisymmetric modes the axisymmetric run cannot represent, domain geometry (cell scale set by box rather than core), or never (mixing always wins, h = 0 is forced)?

Dropped-term ledger (OPS, A limit is sealed with its approach; 2026-10-23)

• The real system: an incompressible fluid at finite viscosity, in finite resolution.
• The ideal: the axisymmetric equations in Hou–Li variables on an annulus that is periodic in z.
• Dropped:
• asymmetric perturbations;
• the time step's own stability (run 027's dt defect, which breaks the CFL bound by 3.3× at a 3× rise in swirl);
• grid resolution at the swirl maximum.
• Here the dropped term is the subject: the viscous term ν∆u is what separates Navier–Stokes from Euler. A blow-up question is a question about whether that term wins.
• What the first correction would call for: dynamically rescaled variables (ξ = r/ℓ(t)), in which a candidate blow-up is a steady state (thread item 34).