39-EFT.WP.Plasma.Confinement v1.0 | Chapter 4 — Minimal Equations: Single-/Two-Fluid & Closures (S20-)

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Chapter 4 — Minimal Equations: Single-/Two-Fluid & Closures (S20-)

I. Chapter Objectives & Structure

1. Objective: Under a releasable unified dialect, provide the single-fluid MHD and two-fluid/two-temperature minimal equation sets (S20-PC-*) for plasma confinement, with explicit sources/sinks and boundaries, conservation laws and energy theorem, anisotropic transport closures and validity ranges—forming testable foundations for equilibrium/stability/waves & transport chapters.
2. Structure: Symbols & domain → Single-fluid minimal set → Two-fluid/two-temperature minimal set → Conservation laws & energy theorem → Closures & sources → Applicability & breakdown → Implementation & records → Falsifiability → Cross-chapter closure.
3. Shared arrival-time dialect (equivalent; 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

• Fields: B(x,t), E(x,t), J(x,t); fluids: n_s(x,t), u_s(x,t), T_s(x,t), p_s = n_s k_B T_s, s∈{e,i}.
• Constants & operators: μ0, ε0, q_s, m_s, ∇·, ∇×, flux-surface average ⟨·⟩_psi.
• Parallel/perpendicular split to B: components (\cdot)_∥ / (\cdot)_⊥.
• Sources & transport: S_n, S_p, Q_s, anisotropic coefficients χ_∥, χ_⊥, resistivity η_∥, η_⊥, viscous tensor Π.
• Regions & interfaces: Ω_core / Ω_edge / Ω_SOL; jump operator [[·]]_Σ.

S20-PC-1 | Minimal Single-Fluid MHD

Conservation & constitutive (dissipative minimal form)

• Mass: ∂ρ/∂t + ∇·( ρ u ) = S_n
• Momentum: ρ ( ∂u/∂t + (u·∇)u ) = J×B − ∇p + ∇·Π + S_m
• Total energy:
  ∂/∂t( ε + B^2/2μ0 ) + ∇·( (ε+p)u + (E×B)/μ0 + Π·u + q ) = P_in − P_rad − P_wall
• Induction: ∂B/∂t = − ∇×E with ∇·B = 0
• Ohm’s law (anisotropic minimal): E + u×B = η_∥ J_∥ + η_⊥ J_⊥
• Ampère & current (quasi-static): μ0 J = ∇×B

Closures & heat flux

• Equation of state: p = n k_B T (single-T) or p = p_e + p_i (two-T total pressure).
• Anisotropic heat flux: q = − χ_∥ ∇_∥ T − χ_⊥ ∇_⊥ T
• Viscosity: Newtonian minimal tensor Π = μ ( ∇u + (∇u)^T − (2/3)I ∇·u ) (omit terms only if documented by mismatch gates).

S20-PC-2 | Minimal Two-Fluid / Two-Temperature

Mass & momentum (per species)

• Mass: ∂n_s/∂t + ∇·( n_s u_s ) = S_{n,s}
• Momentum:
  m_s n_s ( ∂u_s/∂t + (u_s·∇)u_s ) = q_s n_s ( E + u_s×B ) − ∇p_s + ∇·Π_s + R_s
  with R_e = − R_i (inter-species momentum exchange); Π_s anisotropic.

Energy (per species)

• Electrons:
  (3/2) ∂(n_e T_e)/∂t + (3/2) ∇·(n_e T_e u_e) = − p_e ∇·u_e − ∇·q_e + Q_e + C_{ei}
• Ions:
  (3/2) ∂(n_i T_i)/∂t + (3/2) ∇·(n_i T_i u_i) = − p_i ∇·u_i − ∇·q_i + Q_i − C_{ei}
• Heat flux: q_s = − χ_{s,∥} ∇_∥ T_s − χ_{s,⊥} ∇_⊥ T_s

Generalized Ohm’s law (minimal)
E + u_i×B = η J + ( J×B )/(n_e q_e) − ( ∇p_e )/(n_e q_e) − ( m_e / q_e ) ( ∂J/∂t + … )
Hall and electron-pressure-gradient terms are essential; electron inertia term is band-limited.

Couplings

• Quasi-neutrality: ∑_s q_s n_s ≈ 0; current: J = ∑_s q_s n_s u_s.

S20-PC-3 | Conservation Laws & Energy Theorem (release dialect)

• Magnetic energy & Poynting flux:
  ∂(B^2/2μ0)/∂t + ∇·(E×B/μ0) = − E·J
• Volume energy closure: summing fluid energy with the above yields RHS terms for sources/deposition/radiation/wall power—serving as hard gates for check_dim and energy closure.
• Integral form: for any V⊂Ω, the rate of total energy change equals boundary fluxes—recorded as QA.

S20-PC-4 | Closures, Sources & Boundary Conditions (minimal)

• Closures: χ_∥(ψ), χ_⊥(ψ) (radially/flux-surface varying), η_∥, η_⊥, minimal anisotropic Π_s; optional body force S_m and particle/energy sources S_n, Q_s.
• Sources/sinks: P_in = P_RF + P_NBI + …, P_rad (line/volume radiation), P_wall (wall work); all fixed as dataset-card fields.
• Boundaries: at Ω_edge/Ω_SOL impose sheath-compatible conditions (matched particle/energy/momentum fluxes); magnetic boundary & external coil coupling via I10/I20 binding.

S20-PC-5 | Applicability & Breakdown

• Single-fluid MHD: strongly magnetized, low-frequency, homogenized regimes; breaks near ρ_L~L_B, with strong anisotropic heat/electric transport imbalance, or at high frequency.
• Two-fluid/two-T: when Hall and ∇p_e are significant, or at sub-ps/sub-mm or high-frequency propagation; in strongly collisional limits reduces to single-fluid.
• Edge/stochastic layers: islands/stochasticity & steep gradients defer to edge models (Chapter 10).

VI. Implementation & Records (minimum execution dialect)

• Mandatory records:
  model:{single|two-fluid}, closure:{chi_par,chi_perp,eta,viscosity}, sources:{S_n,S_m,Q_s,P_in,P_rad}, BC:{sheath,magnetic}, equilibrium:{psi,q(psi)},
  arrival{form,gamma,measure,c_ref,Tarr,u_Tarr,delta_form}, qa_gates:{check_dim,conservation,stability}.
• Record template:
• s20_model:
• type: "two-fluid"
• closure:
• chi_par_m2s: 5.0
• chi_perp_m2s: 0.5
• eta_Ohm_ohm_m: 1.0e-6
• viscosity: {mu_Pa_s: 1.0e-3}
• sources:
• S_n_m3s: "table:/src/Sn.tbl"
• Qe_Wm3: "table:/src/Qe.tbl"
• Qi_Wm3: "table:/src/Qi.tbl"
• P_in_W: 1.2e6
• P_rad_W: "diag:bolometer"
• bc:
• sheath: "compatible"
• magnetic: {type:"Dirichlet", psi_bc:"/geo/psi_bc.nc"}
• equilibrium: {psi:"/eq/psi.nc", q:"/eq/q.nc"}
• arrival:
• form: "n_over_c"
• gamma: "explicit"
• measure: "d_ell"
• c_ref: 299792458.0
• Tarr_s: 2.31e-06
• u_Tarr_s: 8.0e-08
• delta_form: "n_over_c"
• qa_gates: {check_dim:"pass", conservation:"pass", stability:"pass"}

VII. Falsifiability (for S20-)

• J-S20-1 (Energy non-closure): volume energy vs boundary flux mismatch beyond u → reject current sources/closures.
• J-S20-2 (Anisotropy failure): measured χ_⊥ ~ χ_∥ and weak shear dependence → reject anisotropic closure settings.
• J-S20-3 (Ohm/Hall mismatch): spectral/phase diagnostics disagree with generalized Ohm predictions → reject resistive/Hall parameters.
• J-S20-4 (Two-fluid degenerate inconsistency): in strong-collision limit does not reduce to single-fluid → reject two-fluid parametrization.
• J-S20-5 (Two-dialect T_arr): |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 3 (Postulates & Physical Picture); align with Chapter 5 (Equilibrium/Topology) and Chapter 6 (Stability).
• Downstream: Chapter 7 (Waves/Resonance/Heating, linking n_eff/k_∥/k_⊥ with T_arr/α_abs/P_dep), Chapter 9 (Transport closures & scalings), Chapter 12 (Diagnostics chain), Chapter 13 (Experimental Design & Falsification), Chapter 14 (SimStack & benchmarks).