39-EFT.WP.Plasma.Confinement v1.0 | Chapter 9 — Turbulence, Transport & Closure Models (S70-)

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Chapter 9 — Turbulence, Transport & Closure Models (S70-)

I. Chapter Objectives & Structure

1. Objective: On Chapters 4 (S20-) and 5/8 (S30-/S60-), provide an integrated dialect S70-PC-* for multiscale coupling → quasilinear fluxes → empirical/theoretical closures → flux-surface averaging, unifying publication & QA gates for Γ_s/Q_s/Π_ψ, so turbulent + neoclassical fluxes are computable, regressable, and falsifiable for engineering use.
2. Structure: Symbols & domain → S70-PC-1 multiscale coupling & flux definitions → S70-PC-2 quasilinear & mixing-length closures → S70-PC-3 scaling laws & tabulated closures → S70-PC-4 flux-surface averaging & global balance → S70-PC-5 diagnostic mapping & uncertainty → Implementation & records → Falsifiability → Cross-chapter closure.
3. Shared arrival-time dialect (two equivalent forms; explicit gamma(ell) & d ell; record delta_form):
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )

II. Symbols & Domain

• Basics: n_s,T_s, pressure p_s, mean flows U_∥, U_E; radial coordinate ψ and metric V'(ψ).
• Fluxes & stresses: particle Γ_s, heat Q_s, radial momentum Π_ψ; turbulence intensity I_t, correlation length ℓ_c, time τ_c.
• Dimensionless/scales: ρ_* = ρ_s/a, β, ν_*, gradients a/L_{n,T,p}, safety factor q, small parameter ε=r/R_0.
• Spectra & operators: turbulence spectrum S_k(omega), flux-surface average ⟨·⟩_ψ, time/volume averages ⟨·⟩_t,⟨·⟩_V.

S70-PC-1 | Multiscale Coupling & Flux Definitions (Minimal)

• Volume to flux-surface conservation (particle/energy/momentum):
• ∂_t ⟨ n_s ⟩_ψ + (1/V') ∂_ψ [ V' ⟨ Γ_s ⟩_ψ ] = ⟨ S_{n,s} ⟩_ψ
• ∂_t ⟨ (3/2) n_s T_s ⟩_ψ + (1/V') ∂_ψ [ V' ⟨ Q_s ⟩_ψ ] = ⟨ P_s - C_s ⟩_ψ
• ∂_t ⟨ L_∥ ⟩_ψ + (1/V') ∂_ψ [ V' ⟨ Π_ψ ⟩_ψ ] = ⟨ τ_{ext} ⟩_ψ
• Flux decomposition: Γ_s = Γ_{nc} + Γ_{turb}, Q_s = Q_{nc} + Q_{turb}, Π_ψ = Π_{nc} + Π_{turb} (add S60- neoclassical terms).

S70-PC-2 | Quasilinear & Mixing-Length Closures (QL/ML)

• Quasilinear (minimal):
• Γ_{turb,s} ≈ - D_{QL,s} ∂_ψ n_s + V_{pinch,s} n_s
• , Q_{turb,s} ≈ - χ_{QL,s} ∂_ψ T_s + C_{TE,s} T_s
• Mixing-length:
• D_{QL,s} ~ χ_{QL,s} ~ γ_k / k_⊥^2

where γ_k comes from most-unstable growth rates (see Chapter 6 spectral mapping).

• Cross-terms & thermo-electric coefficients tabulated:
  {V_{pinch}, C_{TE}} = 𝔉( q, ŝ, a/L_{n,T}, ρ_*, β ).

S70-PC-3 | Scaling Laws & Tabulation

• Gyro-Bohm baseline:
• D_GB = ρ_s^2 c_s / a
• , χ_GB ~ C_GB(geometry) · D_GB
• Hybrid empirical/theory closures (examples):
• D_{turb,s} = f_D(q, ŝ, ρ_*, β, a/L_n, a/L_T, ν_*)
• , χ_{turb,s} = f_χ(q, ŝ, ρ_*, β, a/L_T, …)
• Tabulated interface:
  closure_table:{inputs:[q,ŝ,ρ_*,β,a/L_n,a/L_T,ν_*], outputs:[D,χ,V_{pinch},C_{TE}]}; interpolation & extrapolation must be band-limited and QA-annotated.

S70-PC-4 | Flux-Surface Averaging & Global Balance

• Consistent grid & metric: all ⟨·⟩_ψ and ∂_ψ use S30-’s V'(ψ) and metric.
• Global energy closure (hard gate):
• ∑_s ∫ Q_s dψ + P_wall + P_rad = P_in

Volume conservation with boundary powers must close within uncertainty.

S70-PC-5 | Diagnostic Mapping & Uncertainty (M10 anchor)

• Observable mapping: {Γ_s,Q_s,Π_ψ} → {D_α, calorimeters, shear spectra, reflectometry jumps}; spectral windows, sampling, and calibration factors go into dataset cards.
• Uncertainty composition: u(Γ,Q,Π) = 𝒞[ u(S_k), u(grad), u(metrics) ]; report u vs gate thresholds.

VI. Implementation & Records (minimum execution dialect)

• Required fields:
  closure:{type:QL|GB|tabulated, inputs, outputs}, profiles:{n_s(ψ),T_s(ψ)}, geometry:{q(ψ),ŝ(ψ),V'(ψ)}, scales:{ρ_*,β,ν_*},
  fluxes:{Γ_s(ψ),Q_s(ψ),Π_ψ(ψ)}, power_balance:{P_in,P_rad,P_wall}, diagnostics:{windows,cal},
  qa_gates:{check_dim, energy_closure, interpolation_bounds}.
• Record template:
• s70_transport:
• closure:
• type: "tabulated"
• table: "/closures/ql_mix.tbl"
• inputs: [q, s_hat, rho_star, beta, a_over_Ln, a_over_LT, nu_star]
• outputs: [D_turb, chi_turb, V_pinch, C_TE]
• profiles:
• n_e_m3: "grid:/profiles/ne.nc"
• Te_eV: "grid:/profiles/Te.nc"
• geometry:
• q_of_psi: "/eq/q.nc"
• s_hat: "/eq/shat.nc"
• Vprime: "/eq/Vprime.nc"
• scales: {rho_star: 0.003, beta: 0.02, nu_star: 0.08}
• fluxes:
• Gamma_e_m2s: "grid:/flux/Gamma_e.nc"
• Qe_Wm2: "grid:/flux/Qe.nc"
• Pi_Nm: "grid:/flux/Pi.nc"
• power_balance:
• Pin_W: 1.2e6
• Prad_W: 1.5e5
• Pwall_W:1.0e5
• diagnostics:
• windows: {reflectometry_GHz:[...], heat_flux_kHz:[...]}
• cal: "/diag/calibration.yaml"
• qa_gates: {check_dim:"pass", energy_closure:"pass", interpolation_bounds:"pass"}

VII. Falsifiability (for S70-)

• J-S70-1 (Energy/Power non-closure): ∑_s ∫ Q_s dψ + P_wall + P_rad ≠ P_in beyond u → reject closure or power dialect.
• J-S70-2 (Scaling deviation): log-regression residuals of D,χ vs D_GB or target scalings exceed gates → reject scaling or input profiles.
• J-S70-3 (Interpolation out-of-bounds): tabulated closure extrapolation unband-limited or QA-failed → no release.
• J-S70-4 (Spectrum–flux inconsistency): unstable bands in S_k not reflected in inferred D,χ → reject QL/mixing-length assumptions.
• J-S70-5 (Geometry inconsistency): flux-surface averaging inconsistent with S30- metrics (V'(ψ)/grid mismatch) → reject mapping or geometry inputs.

VIII. Cross-Chapter Links & Closure

• Dependencies: Chapter 2 (Terms & Symbols), Chapter 4 (Minimal Equations S20-), Chapter 5 (Equilibrium & Topology S30-), Chapter 6 (Stability & spectra), Chapter 8 (Guiding-center & neoclassical).
• Downstream: Chapter 12 (Diagnostics chain—flux inversion & uncertainty), Chapter 13 (Experimental Design & Falsification—thresholds & power analysis), Chapter 14 (SimStack & benchmarks—flux regression & release gates).