09-EFT.WP.Core.Density v1.0 | Chapter 2 — Physical Density and Conservation

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Chapter 2 — Physical Density and Conservation

I. Scope and Objectives

• Establish a unified convention for physical density rho(x,t), flux J(x,t), and source/sink s(x,t); present integral and differential conservation laws, boundary conditions, and discretization rules; and deliver the mass-balance checklist and workflow Mx-91 required for publication and audit.
• Release the minimal equations S92-1, S92-2, together with norms for control volumes, moving boundaries, finite-volume discretization, and error metrics. Cross-volume anchors align with Core.Sea Chapters 4/6/8.

II. Dimensions and Units (inherits P91-2)

• Dimensions and units.
  unit(rho) = quantity / volume; unit(J) = quantity / ( area * time ); unit(s) = quantity / ( volume * time ).
  Total quantity M(t) = ( ∫_V rho(x,t) dV ), with unit(M) = quantity.
• Audit requirements.
  Publications must provide unit(rho), unit(J), unit(s) and dim(•); all expressions must pass check_dim(expr).

III. Conservation Laws: Integral and Differential Forms

• Minimal equation S92-1 (continuity, differential).
  S92-1 : ∂_t rho(x,t) + ∇•J(x,t) = s(x,t).
• Integral form (fixed control volume V).
  d/dt ( ∫_V rho dV ) + ( ∫_{∂V} J • n dS ) = ( ∫_V s dV ).
  With total M(t) = ( ∫_V rho dV ), net boundary flux Phi_b = ( ∫_{∂V} J • n dS ), and volumetric source S_V = ( ∫_V s dV ),
  dM/dt + Phi_b = S_V.
• Physical decomposition (optional).
  Advective flux J_adv = rho * v; diffusive flux J_diff = -D ∇rho; in general, write J = J_adv + J_diff + J_other, with units provided term by term.

IV. Moving Control Volumes and Relative Flux

V. Boundary Conditions and Conventions

• Dirichlet (value boundary). rho|_{∂V} = rho_D(x,t); used for imposed fields or reservoir interfaces.
• Neumann (flux boundary). ( J • n )|_{∂V} = q_N(x,t); adiabatic/sealed boundary uses q_N = 0.
• Robin (mixed boundary). ( a * rho + b * ( J • n ) )|_{∂V} = r(x,t); used for exchange/radiation boundaries.
• Periodic boundary. On paired faces ∂V_a and ∂V_b, both rho and ( J • n ) match in phase and magnitude.
• Publication requirement. Manifests must list boundary = {"type": "D|N|R|periodic", "params": ...} and specify the alignment of ts and tau_mono.

VI. Source/Sink Modeling and Singular Terms

• Equivalent volumetric source. With volumetric s_bulk(x,t), surface s_surf(x,t) * δ_S, and point sources s_point(t) * δ(x - x0),
  ( ∫_V s dV ) = ( ∫_V s_bulk dV ) + ( ∫_{∂V} s_surf dS ) + ∑_k s_point^{(k)}(t).
• Discretization. Integrate singular sources over voxels to finite values, preserving unit(s); specify time interpolation and alignment to ts.

VII. Conserved Quantities and Audit Metrics

• Minimal equation S92-2 (total and conservation residual).
  S92-2 : M(t) = ( ∫_V rho dV ),
  res_mass(t) = | dM/dt - ( ∫_V s dV ) + ( ∫_{∂V} ( - J • n ) dS ) |.
  Publication threshold: res_mass(t) <= eps_mass, with eps_mass recorded in the manifest.
• Interval conservation. Over [t0,t1]:
  M(t1) - M(t0) = ( ∫_{t0}^{t1} ( ∫_V s dV ) dt ) - ( ∫_{t0}^{t1} ( ∫_{∂V} J • n dS ) dt ).
  Discrete implementations use trapezoidal or Simpson rules in time, reporting the integration order.

VIII. Finite-Volume Discretization and Update Laws

• Voxels and conservation. Grid {V_i}, faces {F_{i→j}}, outward normals. Voxel total M_i^k = rho_i^k * V_i, timestep Delta_t.
• Fluxes and explicit first-order update.
  M_i^{k+1} = M_i^k + Delta_t * ( S_i^k - ∑_{f ∈ ∂V_i} Phi_f^k ),
  with Phi_f^k = ( ∫_{F_f} J^k • n_f dS ), S_i^k = ( ∫_{V_i} s^k dV ).
  Conservation residual:
  res_i^{k+1} = | ( M_i^{k+1} - M_i^k ) / Delta_t - S_i^k + ∑_{f} Phi_f^k |.
• Stability and CFL indicators (recommended constraints).
  Advection: C_adv = max_i( ||v_i|| * Delta_t / Delta_x_i ) <= C_max.
  Diffusion: C_diff = max_i( D_i * Delta_t / Delta_x_i^2 ) <= C'_max.
  Choose C_max, C'_max in the manifest and cite empirical/theoretical sources.

IX. Mass-Balance Workflow Mx-91 (Conservation Checker)

• Load rho/J/s, domain V, and boundary conditions; confirm unit(•) and dim(•) (inherits P91-2).
• Compute M(t_k) = ( ∫_V rho(x,t_k) dV ), estimate dM/dt (difference or smoothed derivative; report window Delta_t).
• Evaluate boundary flux Phi_b(t_k) = ( ∫_{∂V} J • n dS ) and volumetric source S_V(t_k) = ( ∫_V s dV ).
• Form res_mass(t_k) = | dM/dt + Phi_b - S_V | and compare to eps_mass; if exceeded, localize dominant contributions (sensitivity normalized by term).
• Emit an audit report {M, dM/dt, Phi_b, S_V, res_mass, eps_mass, method, ENBW_Hz/Delta_t, ts/tau_mono}; store in the manifest and tag Mx-91-pass|fail.

X. Arrival Time and Timeline Alignment (Cross-Volume Anchors)

• If timing depends on arrival time, report both forms
  Constant pulled out: T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell );
  General: T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ).
  Discrepancy: delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |.
• Audit rule. Any alignment of rho(x,t), J(x,t), s(x,t) with ts that leverages T_arr must record gamma(ell)/d ell/c_ref/n_eff/delta_form in the Mx-91 report (aligned with Core.Sea Chapter 8).

XI. Numerical and Metrological Uncertainty

• Voxel totals.
  u(M_i) ≈ sqrt( ( V_i * u(rho_i) )^2 + ( rho_i * u(V_i) )^2 ).
  For M = ( ∑_i M_i ),
  u(M) = sqrt( ∑_i u(M_i)^2 ) (approximate independence).
• Conservation residual.
  u(res_mass) ≈ sqrt( u(dM/dt)^2 + u(Phi_b)^2 + u(S_V)^2 ); publish expanded uncertainty U = k * u_c and state k.
• Frequency-domain estimate (optional).
  If M(t) is whitened, u(dM/dt) depends on ENBW_Hz; report window power U_w and ENBW_Hz (per Core.Sea Chapter 5).

XII. Minimal Manifest Fields

• domain = {"V":"...", "coords":"...", "dV":"..."}
• boundary = [{"type":"D|N|R|periodic", "params":{...}}]
• sources = {"bulk":"...", "surface":"...", "point":[...]}
• conservation = {"eps_mass":value, "method":"finite-volume|spectral", "time_window":Delta_t}
• timing = {"ts":"UTC", "tau_mono":"...", "T_arr":{"gamma":"...", "d ell":"...", "c_ref":..., "n_eff":"...", "delta_form":...} }
• units = {"rho":"...", "J":"...", "s":"...", "M":"..."}

XIII. Implementation Hooks (I90 excerpts) and Usage

• conserve_mass(dens:DensRef, flow:any, source:any|None=None) -> DensRef
  Enforce S92-1/S92-2 by projecting rho and J to a consistent, conservative state.
• bind_to_equations(eqn_refs:list[str]) -> bool
  Bind ["S92-1","S92-2"] so manifests automatically carry conservation status.
• spectral_density(sig:any, method:str="welch", window:str="hann") -> SpecRef
  Estimate the noise spectrum of dM/dt and compute u(dM/dt) (report with ENBW_Hz).

XIV. Canonical Use Cases and Publishing Notes

• Sealed vessel (J•n = 0). Expect dM/dt = ( ∫_V s dV ); if s = 0, then M(t) is approximately constant and res_mass reflects numerical error and metrological noise.
• Steady outflow (J•n < 0, constant). Steady-state test: | dM/dt | << | Phi_b | and Phi_b ≈ S_V.
• Point injection. At release, accumulate s_point into the interval quantity Q = ( ∫ s_point dt ), and distribute conservatively over voxels.

XV. Chapter Outputs

• Minimal equations defined: S92-1, S92-2.
• Conservation checker workflow provided: Mx-91.
• Boundary types clarified; moving boundary convention; finite-volume update and stability indicators; arrival-time alignment requirements; and uncertainty publication rules formalized.
• Later references: Chapter 7 uses voxel conservation mass_preserve = ( ∑ rho_i * V_i ); Chapter 9 preserves M under cross-domain normalization and records delta_form.