39-EFT.WP.Plasma.Confinement v1.0 | Appendix A. Symbols & Units

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Appendix A. Symbols & Units

I. Conventions & Dialect (Mandatory)

All symbols/formulae appear in English and are wrapped in backticks (e.g., B(x,t), q(psi), n_eff(omega), P_dep(r)).
Any expression with division/integrals/composed operators must use parentheses and explicitly declare the path gamma(ell) and measure d ell.
Arrival-time (two equivalent dialects; record delta_form):
- Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
- General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
Use omega as the frequency variable; if f_Hz is used, state omega = 2π f_Hz.
SI units throughout; key equalities must pass check_dim = pass; cross-chapter hard gates: power closure, topology consistency, stability gates, two-dialect T_arr agreement.
Prohibited: mixing phase/amplitude units (e.g., rad with dB) for the same quantity.

II. Observables & Derived Quantities

Symbol	Meaning	Type/Domain	SI Unit	Notes
B(x,t) / E(x,t)	magnetic/electric field	vector field	T / V·m⁻¹	∇·B=0
J(x,t)	current density	vector field	A·m⁻²	μ0 J = ∇×B (quasi-static)
n_e,n_i; T_e,T_i; p	density/temperature/pressure	scalar field	m⁻³ / eV or K / Pa	p = n k_B T
β	plasma beta	scalar	—	β = 2μ0 p / B^2
psi(R,Z)	poloidal flux	scalar	Wb/2π	S30-
q(psi)	safety factor	scalar	—	topology indicator
Ω_c, ρ_L	cyclotron freq./radius	scalar	s⁻¹ / m	species dependent
v_E, v_{∇B}, v_{curv}	drift velocities	vector	m·s⁻¹	S60-
μ, J_∥	magnetic moment/bounce invariant	scalar	J·T⁻¹ / kg·m·s⁻¹	S60-
n_eff(omega)	effective refractive index	scalar	—	S50-
k_∥, k_⊥	wave-number components	scalar	m⁻¹	related to n_eff
α_abs(ell,omega)	absorption coefficient	scalar	m⁻¹	S50-
P_dep(r)	volumetric deposition	field	W·m⁻³	power closure
Γ_s, Q_s, Π_ψ	particle/heat/momentum flux	scalar	m⁻²·s⁻¹ / W·m⁻² / N·m⁻¹·s⁻¹	S70-
D, χ; D_nc, χ_nc, J_bs	turbulent/neoclassical coeffs.	scalar	m²·s⁻¹ / A·m⁻²	S70-/S60-
q_t(s)	target heat-flux density	scalar	W·m⁻²	S80-
T_arr, T_group	arrival time/group delay	scalar	s	coherence window
Δt_sync	timebase misalignment	scalar	s	Δφ = omega·Δt_sync

III. Path, Arrival & Measure

Symbol	Meaning	Type/Domain	SI Unit	Notes
gamma(ell)	propagation path	geometric curve	m	ray/LOS
d ell	path measure	differential	m	paired with gamma
c_ref	reference speed	constant	m·s⁻¹	arrival baseline
delta_form	dialect tag	enum	—	"n_over_c" / "one_over_c_times_n"

IV. Normalization & Mapping

Symbol	Meaning	Type/Domain	SI Unit	Notes
S(omega)	N-port S-parameters	complex matrix	—	RF chain, if used
Znorm(omega)	port normalization impedance	diagonal/array	Ω	S↔Z baseline
T_mm / Z0_mm	mixed-mode transform/normalization	matrix/diagonal	— / Ω	DM/CM domain
H(x)	observation operator	mapping	—	M10-
R, B	covariances	matrices	—	observation/background

V. Waves, Deposition & Power

Symbol	Meaning	Type/Domain	SI Unit	Notes
D(k,ω)	dispersion matrix	matrix	—	roots set modes
v_g = ∂ω/∂k	group velocity	vector	m·s⁻¹	ray equation
T(ell)	transmission factor	scalar	—	T = exp(-2∫α_abs d ell)
P_in, P_ref, P_rad, P_wall	power balance	scalar	W	closure hard gate

VI. Metrology & Alignment

Symbol	Meaning	Type/Domain	SI Unit	Notes
AF / PF	antenna/probe factor	file/calibration	—	versioned
windows{band,bw,fs}	spectral window/bandwidth/sampling	params	Hz	dataset card
T_b(ω), S_obs(k,ω)	brightness/scattering spectrum	observable	K / per defn.	transfer matrix

VII. Canonical Relations (dialect-aligned, quick ref)

Grad–Shafranov: Δ* psi = − μ0 R^2 p'(psi) − F F'(psi)
Energy principle: δW[ξ] > 0 (stable), tearing Δ' ≤ 0.
Ray equations: dx/ds = ∂ω/∂k, dk/ds = − ∂ω/∂x
Absorption/transmission: dP/dell = − 2 α_abs P, T = exp( − 2 ∫ α_abs d ell )
Deposition closure: ∫_V P_dep dV = P_in − P_ref − P_rad − P_wall
Arrival correction: arg Z_corr(ω) = arg Z_raw(ω) − ( ω · Δt_sync )
Group delay: T_group(ω) = d/dω( arg Z_eft(ω) )
Flux-surface average: ∂_t ⟨ n ⟩_ψ + (1/V') ∂_ψ ( V' ⟨ Γ_ψ ⟩_ψ ) = ⟨ S_n ⟩_ψ
Gyro-Bohm scaling: D_GB = ρ_s^2 c_s / a

VIII. Forbidden Aliases & Usage (Mandatory)

T_fil (tension) ≠ T_trans (transmittance).
n (number density) ≠ n_eff (effective refractive index).
Bare c is forbidden; use c_ref.
Ambiguous sigma: use sigma_cond for conductivity, sigma_std for standard deviation.
Frequency variable is uniformly omega; f only in prose, never in equations.
Do not mix rad and dB in the same equation.
Arrival expressions must annotate gamma(ell) and d ell, and record delta_form.

IX. Writing Examples (compliance snippets)

Arrival record:
arrival:
form: "n_over_c" # or "one_over_c_times_n"
gamma: "explicit"
measure: "d_ell"
c_ref: 299792458.0
Tarr_s: 3.10e-06
u_Tarr_s: 9.0e-08
delta_form: "n_over_c"
Power closure (dataset fragment):
power_balance:
Pin_W: 1.20e6
Pref_W: 2.00e5
Prad_W: 1.50e5
Pwall_W: 1.00e5
Pdep_W: "integrate:/waves/Pdep.nc"

X. SI Quick Reference
Current A; voltage V; impedance Ω; phase rad; time s; length m; mass kg; temperature K; energy J; power W; magnetic flux density T; charge C; frequency Hz; heat-flux density W·m⁻²; diffusion coefficient m²·s⁻¹.

XI. Release Hard Gates (Consistency)

two-dialect T_arr agreement (|Δ| ≤ u(T_arr)).Arrival: δW_min > 0 ∧ Δ' ≤ 0 ∧ {D_S,D_M}>0; Stability: q(ψ)/LCFS/X matches magnetics/imaging; Topology consistency: P_in = P_ref + P_rad + P_wall + ∫ P_dep dV; Power closure:check_dim = pass;

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Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
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Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.

First published： 2025-11-11｜Current version：v5.1
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