39-EFT.WP.Plasma.Confinement v1.0 | Chapter 6 — Stability & Energy Principle (S40-) (S30-)

Home ／ Docs-Technical WhitePaper ／ 39-EFT.WP.Plasma.Confinement v1.0

Chapter 6 — Stability & Energy Principle (S40-) (S30-)

I. Chapter Objectives & Structure

1. Objective: On the established equilibrium background (see Chapter 5, S30-), provide a unified dialect and release criteria for stability assessment covering ideal/resistive MHD modes, small-scale drift-type instabilities, the minimal expression of the energy principle δW, and the engineering workflow for diagnostic frequency-domain mapping—so subsequent waves/heating and transport chapters close under a common stability context.
2. Structure: Symbols & domain → S40-PC-1 ideal/resistive MHD modes → S40-PC-2 small-scale instabilities → S40-PC-3 energy principle & criteria → S40-PC-4 spectral mapping & diagnostics → S40-PC-5 stability boundaries & operating window → 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

• Equilibrium & geometry: psi(R,Z), q(psi), B = B_θ e_θ + B_φ e_φ, curvature κ = b·∇b, flux-surface average ⟨·⟩_psi.
• Perturbations & spectra: displacement ξ(x,t), perturbations δB, δp, δJ, mode labels (n,m), eigenfrequency ω, growth rate γ = Im{ω}.
• Energy functional & operators: δW[ξ], parallel/perpendicular splits (\cdot)_∥ / (\cdot)_⊥, jump operator [[·]]_Σ (tearing layer).
• Dissipation & closures: resistivity η, viscosity Π, anisotropic heat conductivities χ_∥/χ_⊥.
• Diagnostics: magnetics/ECE/soft-X/reflectometry/MSE and their frequency/phase dialects.

S40-PC-1 | Ideal/Resistive MHD Modes (Minimal)

Ideal MHD linearization & eigenproblem

• Linear Euler–Lagrange form: ρ ∂^2 ξ/∂t^2 = 𝔽(ξ), reducing stability to the positive-definiteness of the energy functional δW[ξ].
• External modes (kink / external tearing precursors): spectra set by boundary displacement and current-closure conditions; internal modes (sawtooth precursors) emerge near q ≈ 1.

Resistive effects & tearing layer

• In resistive MHD, an inner layer at q = m/n is matched by the parameter
  Δ' = [ (d ln ψ̃ / dr) ]_{out}.
  Release gates: stable if Δ' < 0 (per this volume’s dialect); unstable if Δ' > 0—pair with mitigation strategy and thresholds.

S40-PC-2 | Small-Scale Instabilities (Drift / Skin / Tearing extensions)

• Drift-type (ITG/ETG/TE, etc.): growth driven by free-energy channels (density/temperature gradients) coupled to magnetic shear; publish as threshold bands and, in engineering dialect, as sources for anisotropic diffusion closures (linked to Chapter 9).
• Skin (resistive-skin) instabilities: coupling of resistive layers and geometric shear; enhanced at high η and strong ∂J_∥/∂r; publish as banded gate on frequency/gradient windows.
• Tearing extensions: include fluid–kinetic corrections (thermal conduction, inertia, Hall); inner-layer matching must preserve δW energy consistency.

S40-PC-3 | Energy Principle & Criteria (release dialect)

Energy functional (minimal expression)

δW = (1/2) ∫_V { (B^2/μ0) |∇⊥·ξ|^2 + γ_p p (∇·ξ)^2

+ (ξ·∇p)(ξ·κ) + |δB|^2/μ0 } dV + 𝔅[ξ]_boundary

Require for all admissible ξ that δW > 0 (Newcomb stability). Boundary term [ξ]_boundary comes from external vacuum and coils (see S30-/I20-).

Local criteria (examples)

• Suydam / interchange: combined pressure-gradient and magnetic-shear indicator D_S(ψ); publish gate D_S > 0 (stable).
• Mercier: curvature/profile indicator D_M(ψ); publish gate D_M > 0.
• Tearing energy gate: require both Δ' and δW gates to pass—avoid single-indicator releases.

S40-PC-4 | Spectral Mapping & Diagnostics (M10 anchor)

• Eigen- to diagnostic-spectra: map ω(n,m) and γ(n,m) to feature peaks/phase jumps in magnetics/ECE/soft-X channels; record spectral windows & bandwidths in dataset cards.
• Phase & arrival correction: within the coherence window apply
  arg Z_corr(omega) = arg Z_raw(omega) − ( omega · Δt_sync )
  before linear or eigen-spectrum fitting (consistent with Chapter 7 arrival dialect).
• Stability boundary regression: regress operation windows using δW/Δ' against measured spectral/phase features.

S40-PC-5 | Stability Boundaries & Operating Window (Engineered Gates)

• Core gates: δW_min > 0, Δ' ≤ 0, and local indicators {D_S, D_M} > 0.
• Secondary gates: growth-rate cap γ ≤ γ_gate, or feature amplitude A_peak ≤ A_gate.
• Transfer strategies: if gates fail, emit a geometry/profile remedy table (increase shear, relax gradients, external coil settings) and record pre/post-action consistency gates.

VI. Implementation & Records (minimum execution dialect)

• Required fields:
  equilibrium:{psi,q(psi)}, geometry:{LCFS,Xpoints,coils}, energy_principle:{δW_min, boundary_term},
  tearing:{Δ', layer_model}, local_criteria:{D_S(ψ), D_M(ψ)}, spectrum:{ω(n,m), γ(n,m)},
  diagnostics_map:{magnetics,ECE,SXR}, qa_gates:{check_dim, energy_closure, topology}.
• Record template:
• s40_stability:
• equilibrium: {psi:"/eq/psi.nc", q:"/eq/q.nc"}
• energy_principle:
• deltaW_min_J: 2.3e3
• boundary_term: "vac_coupled"
• tearing:
• DeltaPrime_m1: -0.8
• layer_model: "resistive_MHD"
• local_criteria:
• D_S: "profile:/crit/DS.tbl" # > 0 -> stable
• D_M: "profile:/crit/DM.tbl" # > 0 -> stable
• spectrum:
• modes:
• - {n:1, m:1, omega_rad_s: 1.2e5, gamma_rad_s: 2.0e3}
• - {n:2, m:1, omega_rad_s: 2.0e5, gamma_rad_s: 0.0}
• diagnostics_map:
• magnetics: "/diag/mag.cfg"
• ECE: "/diag/ece.cfg"
• SXR: "/diag/sxr.cfg"
• qa_gates: {check_dim:"pass", energy_closure:"pass", topology:"pass"}

VII. Falsifiability (for S40-)

• J-S40-1 (Negative δW): δW_min ≤ 0 → reject current equilibrium/profile settings or boundary-term dialect.
• J-S40-2 (Δ' > 0 with spectral match): Δ' > 0 and diagnostics show corresponding tearing peaks/phase features → unstable; if mitigation fails to reach Δ' ≤ 0, no release.
• J-S40-3 (Local-criterion conflict): D_S or D_M fails while eigen-spectrum shows no instability (or vice versa) → roll back profiles/geometry or layer model; unresolved conflict → no release.
• J-S40-4 (Energy closure mismatch): δW + ∫_V E·J dV not closing with boundary Poynting flux → reject energy-principle implementation or diagnostic mapping.
• J-S40-5 (Two-dialect arrival): |T_arr^{(n_over_c)} − T_arr^{(one_over_c_times_n)}| > u(T_arr) → no release.

VIII. Cross-Chapter Links & Closure

• Dependencies: Chapter 2 (Terms & Symbols), Chapter 5 (Equilibrium & Topology); align with Chapter 4 (Minimal Equations) for energy/Ohm’s dialects.
• Downstream: Chapter 7 (Waves, Resonance & Heating; arrival-phase & spectral fitting), Chapter 9 (Turbulent-transport closure parameterization), Chapter 12 (Diagnostics chain for spectral mapping), Chapter 13 (Experimental Design & Falsification—stability gates).