34-EFT.WP.Astro.Acceleration v1.0 | Chapter 5 Shear Acceleration

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Chapter 5 Shear Acceleration

I. Abstract & Scope
This chapter provides the geometric and statistical packaging of shear fields, establishes the minimal S40-* set driven by the shear strain rate sigma_shear for energy gain and acceleration timescales, and supplies M40-* workflows to interface with spectrum formation (Ch.7) and transport/losses (Ch.8). 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-* (A_acc(E), tau_acc(E), g_cycle(E), tau_cycle(E)).
• Reconnection comparator & composition: Chapter 4 S30-* (A_rec(E)).
• Metrology: EFT.WP.Core.Metrology v1.0 Ch.1–3.

III. Normative Anchors (added in this chapter)

• S40-0 (Geometry & Frames): The shear layer is described by the velocity-gradient tensor S = ( ∇v + ( ∇v )^T ) / 2, principal gradient direction e_grad, velocity contrast Delta_u, gradient scale L_grad, and layer thickness delta_shear. Evaluation is in F_shear and mapped to F_flow.
• S40-1 (Shear Metric): sigma_shear = ( 0.5 * Tr( S^2 ) )^{1/2}; Delta_u ≈ | ( ∂u/∂n ) | * L_grad, with normal n ‖ e_grad.
• S40-2 (Cycle Time): tau_shear(E) = tau_adv_sh + tau_sc_sh(E), where tau_adv_sh ≈ ( L_grad / Delta_u ).
• S40-3 (Per-Cycle Gain Law): g_sh(E) = k_gs * ( sigma_shear * tau_sc_sh(E) )^{p_sh} * Phi_sh, with p_sh ≥ 1, Phi_sh = chi_aniso * f_geom( delta_shear / L_grad ), chi_aniso ∈ (0,1].
• S40-4 (Rate Closure): A_shear(E) = g_sh(E) / tau_shear(E).
• S40-5 (High-Energy Roll-Off): f_coh(E) = ( 1 + ( E / E_cut_sh )^{p_coh} )^{-1}, then A_shear(E) ← A_shear(E) * f_coh(E).
• S40-6 (Laminar/Turbulent Limits): In the laminar limit omega_vec = ∇ × v_vec → 0, set p_sh ≈ 2; in the turbulent limit, replace tau_sc_sh(E) by a correlation time and use the time-mean ⟨ sigma_shear ⟩_t.
• S40-7 (Composition with Total Acceleration): A_acc(E) = A_rec(E) + A_shear(E).

IV. Body Structure

I. Shear-Field Modeling

• Frames: Diagonalize S in F_shear to define e_grad; map F_shear → F_flow for observationally relevant timescales.
• Geometry: L_grad is the characteristic scale of the velocity normal gradient; delta_shear the layer thickness; Delta_u the two-sided velocity contrast.
• Anisotropy: chi_aniso captures scattering anisotropy and Thread/magnetic orientation; Phi_sh aggregates geometric factors.

II. Key Equations & Derivations (S-series)

• S40-1: sigma_shear = ( 0.5 * Tr( S^2 ) )^{1/2}.
• S40-2: tau_shear(E) = ( L_grad / Delta_u ) + tau_sc_sh(E).
• S40-3: g_sh(E) = k_gs * ( sigma_shear * tau_sc_sh(E) )^{p_sh} * chi_aniso * f_geom( delta_shear / L_grad ), with f_geom(0)=0, monotone non-decreasing, f_geom(1)≤1.
• S40-4: A_shear(E) = g_sh(E) / ( ( L_grad / Delta_u ) + tau_sc_sh(E) ).
• S40-5: A_shear(E) ← A_shear(E) * ( 1 + ( E / E_cut_sh )^{p_coh} )^{-1}.
• S40-6 (Laminar limit): If omega_vec → 0 and scattering is isotropic, set p_sh = 2, giving g_sh(E) ∝ ( sigma_shear * tau_sc_sh(E) )^2.
• S40-7 (Turbulent limit): Use correlation time tau_corr in place of tau_sc_sh(E); take sigma_shear as ⟨ sigma_shear ⟩_t.
• S40-8 (Total Acceleration): A_acc(E) = A_rec(E) + A_shear(E); tau_acc(E) = 1 / A_acc(E) for Chapter 7.

III. Methods & Flows (M-series)

• M40-1 (Tensor Estimation & Zoning): From v_vec(x,t) compute S and omega_vec; threshold to segment shear-dominated regions; output {sigma_shear, L_grad, delta_shear, Delta_u} with uncertainties.
• M40-2 (Rates & Timescales): Using S40-2…S40-5, compute {g_sh(E), tau_shear(E), A_shear(E)}; provide energy-binned outputs with priors/posteriors recorded.
• M40-3 (Laminar/Turbulent Decision): If ||omega_vec|| / sigma_shear < k_thresh use laminar closure (p_sh=2); otherwise use turbulent closure and estimate tau_corr.
• M40-4 (Channel Discrimination): Joint fit with Chapter 4 to infer relative weights of {A_rec(E), A_shear(E)}; report evidence ratios and channel posteriors.
• M40-5 (Multiband Fitting): Minimize L = L_spec + L_timing + L_polar jointly over spectrum, timing, and polarization. Any ToA terms must use both forms
  T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) and T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ), explicitly carrying gamma(ell) and d ell, and recording delta_form.
• M40-6 (Sensitivity Analysis): Perturb {chi_aniso, p_sh, E_cut_sh, p_coh} and report sensitivity matrices for {A_shear(E), alpha_spec, Pi}.

IV. Cross-References within/beyond this Volume

• Kinematics: Chapter 3 S20-* (definitions and composition of A_acc, tau_acc).
• Reconnection comparator: Chapter 4 S30-* (A_rec(E)); channel discrimination via M40-4.
• Spectrum formation & transport: Chapter 7 S50-, Chapter 8 S52-.
• ToA & paths: all ToA-related integrals carry gamma(ell) and d ell; both ToA forms are used and delta_form recorded.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • An energy band with tau_shear(E) < tau_loss(E) and spectral hardening.
   • A_shear(E) monotonically increases with sigma_shear and correlates with Delta_u / L_grad.
   • Polarization angle drifts smoothly; polarization fraction increases moderately with stronger shear.
2. Negative criteria:
   • If sigma_shear → 0 or chi_aniso → 0 and fit quality does not degrade, the shear channel is falsified.
   • Abnormal sensitivity of A_shear(E) to F_shear → F_flow mapping indicates geometric or statistical inconsistency.
3. Contrasts: Hold A_loss(E) and A_rec(E) fixed; compare A_shear(E)=0 vs A_shear(E)>0 to localize observational differences.

VI. Summary & Handoff
This chapter delivers the S40-* geometric–statistical packaging and energy-gain closure for shear, plus M40-* computation and fitting flows coupled to Chapters 7–8. Chapter 6 provides the comparator and boundary conditions for shocks and turbulence, constructing equivalence/degeneration relations for channel-level discrimination.

V. Figures & Tables (this chapter)

• Fig. 5-1 Shear-layer geometry and frames (F_shear ↔ F_flow, e_grad, L_grad, delta_shear, Delta_u).
• Tab. 5-1 Local Symbol Table (this chapter)

• Tab. 5-2 Parameters & Default Priors (examples)

• Tab. 5-3 Observational Criteria