34-EFT.WP.Astro.Acceleration v1.0 | Chapter 6 Shocks & Turbulence: Comparator & Boundaries

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Chapter 6 Shocks & Turbulence: Comparator & Boundaries

I. Abstract & Scope
This chapter serves as the comparator/boundary layer. It packages diffusive shock acceleration (DSA) and second-order turbulent acceleration into a unified form, establishes equivalence/degeneration conditions against the reconnection/shear channels, and provides actionable discrimination criteria. Equations use English notation with backticks; SI units are used; composite expressions are parenthesized.

II. Dependencies & References

• Terms & symbols: Chapter 2 Tab. 2-1 and P12-*.
• Kinematic baseline: Chapter 3 S20-*.
• Reconnection & shear channels: Chapter 4 S30-, Chapter 5 S40-.
• Spectrum formation & transport: Chapter 7 S50-, Chapter 8 S52-.
• Inference & falsification: Methods.Inference v1.0, Methods.Falsification v1.0.

III. Normative Anchors (added in this chapter, S45-*)

• S45-0 (Geometry & Frames): For shocks, use upstream/downstream speeds u1/u2, compression ratio r = u1 / u2 > 1, jump Delta_u = u1 - u2, and thickness delta_shock; evaluate in the shock frame and map to F_flow. For turbulence, use turbulent speed u_turb, correlation length L_corr, and correlation time tau_corr; evaluate in a co-moving statistical frame and map to F_flow.
• S45-1 (DSA Cycle Time): tau_dsa(E) = C_dsa * D(E) / ( Delta_u^2 ), with C_dsa > 0 dimensionless and D(E) the diffusion coefficient.
• S45-2 (DSA Per-Cycle Gain): g_dsa(r) = C_g(r), obeying C_g(1)=0, ∂C_g/∂r > 0, and C_g(r) < 1.
• S45-3 (DSA Rate Closure): A_dsa(E) = g_dsa(r) / tau_dsa(E).
• S45-4 (Turbulent Cycle Time): tau_turb(E) = tau_corr + tau_sc_turb(E).
• S45-5 (Turbulent Per-Cycle Gain): g_turb(E) = k2 * ( u_turb / c_ref )^2 * Phi_turb, with k2 > 0 dimensionless and Phi_turb ∈ (0,1] aggregating geometry/anisotropy.
• S45-6 (Turbulent Rate Closure): A_turb(E) = g_turb(E) / tau_turb(E).
• S45-7 (Degeneration to Shear): If delta_shock is finite and sigma_shear ≈ |∂u/∂n| ≈ Delta_u / delta_shock with near-isotropic scattering, there exists an energy band where A_dsa(E) ≈ A_shear(E) (Chapter 5), i.e., a shear-limit degeneration.
• S45-8 (Equivalence to Reconnection): If the shock is energetically fed by sheet reconnection, with u_out ≈ u1 and openness chi_open governing flux, there exists an energy band where A_dsa(E) ≈ A_rec(E) (Chapter 4), i.e., a reconnection-equivalence limit.
• S45-9 (Dominance Factor): Define eta_dom(E) = A_channel(E) / max{ A_other(E) }. If eta_dom(E) ≥ eta_* with threshold eta_* > 1, the channel is dominant in that band.
• S45-10 (Comparator Falsification Line): If replacing A_rec(E)/A_shear(E) by A_dsa(E) or A_turb(E) does not degrade fit quality and the evidence ratio does not worsen, the nominal channel is falsified or nonessential.

IV. Body Structure

I. Comparator Models & Boundaries

• Shock vs shear: with gradients concentrated in a thin layer and finite Delta_u / delta_shock, the effective DSA rate is approximated by the shear closure; in the laminar limit (omega_vec → 0), approach Chapter 5’s laminar regime.
• Shock vs reconnection: when sheet reconnection supplies power and yields macroscopic compression/outflow, the shock geometric flux factor maps to reconnection Phi_geom.
• Turbulence vs shear/reconnection: turbulence manifests as slow variation with random walk in time; energy dependences in g_turb(E) and tau_turb(E) can overlap with shear roll-off or reconnection efficiency, requiring joint evidence and criteria to disentangle.

II. Key Equations & Derivations (S-series)

• S45-1: tau_dsa(E) = C_dsa * D(E) / ( Delta_u^2 ).
• S45-2: g_dsa(r) = C_g(r) with C_g(1)=0, monotone increasing, and < 1.
• S45-3: A_dsa(E) = g_dsa(r) / tau_dsa(E).
• S45-4: tau_turb(E) = tau_corr + tau_sc_turb(E).
• S45-5: g_turb(E) = k2 * ( u_turb / c_ref )^2 * Phi_turb.
• S45-6: A_turb(E) = g_turb(E) / tau_turb(E).
• S45-7 (Degeneration to Shear): A sufficient condition for A_dsa(E) ≈ A_shear(E) is delta_shock ~ L_grad and sigma_shear ≈ Delta_u / delta_shock under Chapter 5’s laminar closure.
• S45-8 (Equivalence to Reconnection): A sufficient condition for A_dsa(E) ≈ A_rec(E) is a combination of u_out / c_ref, chi_open, and Theta_T that aligns reconnection Phi_geom with the shock flux factor.
• S45-9 (Dominance): eta_dom(E) = A_channel(E) / max{ A_rec(E), A_shear(E), A_dsa(E), A_turb(E) } excluding A_channel(E) from the denominator.

III. Methods & Flows (M-series)

• M45-1 (Unified Packaging): In a single data/simulation stack compute {A_rec(E), A_shear(E), A_dsa(E), A_turb(E)} with energy-binned outputs and uncertainties.
• M45-2 (Equivalence/Degeneration Test): For a given energy band, compute ΔL = L_channel − L_alt and evidence ratio K = Z_channel / Z_alt; if ΔL ≥ 0 and K ≤ 1, classify as equivalent or degenerate.
• M45-3 (Dominant-Band Extraction): Using eta_dom(E) and threshold eta_*, mark dominant intervals and output cross-channel masks for use in spectrum formation and transport.
• M45-4 (Comparator Experiment Design): With A_loss(E) fixed, set one of {A_rec, A_shear, A_dsa, A_turb} to zero in turn and compare observables to localize the gain source.
• M45-5 (ToA Terms): All time-of-arrival fits proceed in parallel with both forms, explicitly carrying path and measure and recording delta_form:
  T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ).

IV. Cross-References within/beyond this Volume

• Kinematics: Chapter 3 S20-* (composition rules and timescales).
• Reconnection/Shear: Chapter 4 S30-, Chapter 5 S40- (channel mapping and discrimination).
• Spectrum/Transport: Chapter 7 S50-, Chapter 8 S52- (compute alpha_spec and flux under channel masks).
• Inference/Falsification: Methods.Inference v1.0, Methods.Falsification v1.0 (evidence ratios and power analysis).

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Shock-dominant: A_dsa(E) increases with Delta_u; spectrum hardens where tau_dsa(E) < tau_loss(E) and is sensitive to r; polarization angle often steps.
   • Turbulence-dominant: timing statistics show slow variation with random walk; A_turb(E) scales with ( u_turb / c_ref )^2; polarization coherence decreases.
2. Negative criteria:
   • If Delta_u → 0 or u_turb → 0 and fit quality does not degrade, the respective comparator channel is falsified or nonessential.
   • If setting A_dsa or A_turb to zero yields evidence ratio K ≥ 1, the comparator channel is unsupported by data.
3. Contrasts: With A_loss(E) fixed, compare {only DSA, only turbulence, reconnection+shear, full} configurations and report spectral/timing/polarimetric signatures.

VI. Summary & Handoff
This chapter finalizes comparator packaging for shocks and turbulence, establishes equivalence/degeneration with reconnection/shear, and provides procedural extraction of dominant energy bands. Chapter 7 builds on this to deliver closed or semi-closed solutions for energy gain and spectrum formation with observational mappings.

V. Figures & Tables (this chapter)

• Fig. 6-1 Comparator schematic: F_shock ↔ F_flow, u1/u2/Delta_u, delta_shock, u_turb/L_corr.
• Tab. 6-1 Local Symbol Table (this chapter)

• Tab. 6-2 Channel Discrimination Checklist

• Tab. 6-3 ToA Recording Template