Formula reference

vaft.formula is documented from a single source of truth: every public function carries a standardized docstring (summary, definition, units, convention, assumptions, validity, limitations, numerical notes, references), and these pages are generated from those docstrings by python -m vaft.formula.catalog. Nothing here is written by hand, so the site and vaft.formula.describe("<name>") always agree.

import vaft.formula as F

print(F.describe("greenwald_density"))          # one formula, rendered
F.search("Sauter")                              # every formula whose text mentions it
F.list_formulas(category="stability")           # imports only that submodule
F.describe("greenwald_density").definition      # its $$...$$ equation, from the docstring

In Jupyter the same entry renders as a Markdown card with typeset equations when it is a cell’s last expression, and F.show renders only the parts a notebook needs, in the order given, so a tutorial shows the docstring’s equation instead of restating it:

F.describe("greenwald_density")                 # the whole card
F.show("greenwald_density", sections=["definition", "convention", "validity"])

A part is signature, summary, definition, description, parameters, returns, raises, references or a docstring section (convention, validity, …). Asking for one the formula does not document raises ValueError; print(...) keeps the terminal text.

The discovery layer is loaded on first use only; import vaft.formula.stability never touches it.

Category	Module	Functions	Contents
constants	`vaft.formula.constants`	0	Physical constants and the confinement-scaling coefficient table.
utils	`vaft.formula.utils`	12	Utility functions for plasma physics calculations.
equilibrium	`vaft.formula.equilibrium`	111	Plasma equilibrium, current, energy, and geometry calculations.
virial	`vaft.formula.virial`	33	Virial closures: global quantities from the Shafranov boundary integrals.
stability	`vaft.formula.stability`	45	Plasma stability, operational limits, and transport calculations.
green	`vaft.formula.green`	16	Green's function calculations for plasma physics.
atomic	`vaft.formula.atomic`	11	Atomic equilibrium and line-radiation formulae backed by OPEN-ADAS ADF11.
statistics	`vaft.formula.statistics`	22	Pure statistical kernels for residual and convergence diagnostics (issue #186).
magnetics	`vaft.formula.magnetics`	2	Magnetic-probe geometry shared by every forward model.
neoclassical	`vaft.formula.neoclassical`	18	Analytic neoclassical transport formulas: Sauter and Redl.
startup	`vaft.formula.startup`	32	Start-up physics from the gas fill to the first closed flux surfaces.
transformer	`vaft.formula.transformer`	8	The plasma as a transformer secondary: Romero's exact current and inductance balance.
particle	`vaft.formula.particle`	13	Single-particle motion in magnetic and electric fields.
geometry	`vaft.formula.geometry`	16	Canonical geometric approximations: slab, sheared slab, cylinder, local reduction.
ripple	`vaft.formula.ripple`	6	Toroidal-field ripple from a finite number of TF coils, and its orbit consequences.
disruption	`vaft.formula.disruption`	11	Disruption reference relations: thermal and current quench, induced field, runaway electrons.
nbi	`vaft.formula.nbi`	6	Neutral beam injection: particle rate, neutral attenuation along a path, birth density, shine-through.
waves	`vaft.formula.waves`	7	Cold-plasma waves: characteristic frequencies, Stix parameters, refractive indices, cutoffs and resonances.
ntv	`vaft.formula.ntv`	2	Neoclassical toroidal viscosity: the analytic relations of transport in broken toroidal symmetry.
vde	`vaft.formula.vde`	6	Vertical displacement events: local motion, wall response and halo-current descriptors.
pwi	`vaft.formula.pwi`	4	Plasma-wall interaction: collision kinematics, reflection and recycling definitions, sputtering threshold.
sol	`vaft.formula.sol`	19	Scrape-off layer: sound speed, sheath fluxes, parallel conduction and the target heat-flux profile.
boundaries	`vaft.formula.boundaries`	15	Operational boundaries from the literature: one data model for density, transition and stability limits.
impurity	`vaft.formula.impurity`	9	Impurity-mixture algebra: moments, target-Z_eff densities, the reduced pseudo-impurity and main-ion dilution.
dimensional	`vaft.formula.dimensional`	7	Dimensional analysis: dimension matrices, Buckingham-Pi groups and similarity constraints.
sensitivity	`vaft.formula.sensitivity`	6	Generic kernels for local sensitivity, linear uncertainty propagation and identifiability (issue #1642).
ordering	`vaft.formula.ordering`	16	Asymptotic ordering parameters: the scale separations reduced models assume.
fast_ion	`vaft.formula.fast_ion`	8	Classical fast-ion slowing down: critical velocity, slowing-down time, distribution, density and pressure.
kinetic	`vaft.formula.kinetic`	7	Collisional closures of a multi-species plasma: collision times, energy exchange and classical diffusivities.
turbulence	`vaft.formula.turbulence`	5	Two-variable turbulence--zonal-flow predator--prey model (Lotka--Volterra form).
regularized_profiles	`vaft.formula.regularized_profiles`	3	Positive, regular flux-function profiles; independent of machine and ODS.

Conventions that change the number

Every function whose result depends on a sign, normalisation, COCOS or unit choice carries a Convention section and is marked convention-sensitive on its page; empirical fits open their Validity section with Empirical fit. and name the dataset or publication. The recurring traps:

Poloidal flux, Wb versus Wb/rad. The equilibrium helpers default to flux per radian (COCOS 1-8, EFIT g-files, VFIT); the IMAS Data Dictionary and the Green’s functions use full weber (COCOS 11-18). poloidal_field_factor(cocos) carries both the $2\pi$ and the orientation sign; vaft.data.eqdsk.ods_psi_to_wb_per_radian_factor settles which family an ODS holds – a declared COCOS index first, the data probes only for an unlabelled file. vaft.omas.equilibrium_psi_to_weber moves a legacy Wb/rad ODS to the DD’s weber in place and declares COCOS 11; the packaged 39915 sample is stored that way since issue #478.
Engineering units. The confinement scalings, $n_G$, $\beta_N$ and the Verdoolaege dimensionless parameters are defined in MA, MW, $10^{19}$ m$^{-3}$ and percent; each page states what is converted internally and what is not.
Several definitions of one quantity. Three $\nu_$ and three $\rho_$ definitions coexist (tracked in #353); each page names its own.

Constants

Every physics submodule of vaft.formula takes its constants from here so that a value is changed in one place. The electromagnetic and particle constants are CODATA 2018 recommended values [1] (MU0 keeps the pre-2019 exact definition $4\pi\times10^{-7}$, which differs from the 2018 measured value by $5\times10^{-10}$ relative); E_ALPHA is the D-T alpha birth energy [2]. SIGMA_V_COEF, SPITZER_RESISTIVITY_COEF and COLLISIONALITY_COEF are fitted prefactors whose meaning is defined by the single function that uses each of them, documented there.

The private _SCALING_COEFS table holds the engineering-unit confinement scalings (ITER89P, IPB98(y,2), ITER97-L, the two NSTX 2006 fits, the Kurskiev 2022 ST fit and the ITPA20 / ITPA20-IL H-mode fits of Verdoolaege 2021). Each entry carries its prefactor, exponents, target density definition, source reference and unit-convention notes; the table is read by vaft.formula.equilibrium.confinement_time_from_engineering_parameters, which documents the conventions and cites every entry.

Name	Meaning	Unit
`MU0`	vacuum permeability	H/m
`PA_PER_TORR`	one torr, by definition 101325/760	Pa
`EPS0`	vacuum permittivity	F/m
`QE`	elementary charge	C
`ME`	electron mass	kg
`MI_P`	proton mass	kg
`E_ALPHA`	D-T alpha particle birth energy, 3.5 MeV	J
`SIGMA_V_COEF`	prefactor of the rough D-T reactivity fit 1.1e-24 T_keV^2	m^3/s/keV^2
`SPITZER_RESISTIVITY_COEF`	prefactor of eta = 5.2e-5 Z_eff ln(Lambda) / T_eV^1.5	Ohm m eV^1.5
`COLLISIONALITY_COEF`	prefactor of the IPB98 nu_star = 6.921e-18 n_19 R / (T_keV^2 B_t) form	-

All formulas

Formula Category Summary

calculate_peaking_factor utils Peaking factor, central value over volume average.

calculate_poloidal_flux utils Poloidal flux per radian from a line integral of $RB_\theta$. Convention

calculate_toroidal_flux utils Toroidal flux as a sum of $B_\varphi$ over area elements. Convention

calculate_volume_weighted_average utils Volume-weighted average of a sampled profile.

eped_tanh_bounds utils Parameter box the eped_tanh fit uses for this data.

fit_profile utils Fit a 1-D profile with a selectable model and evaluate it on a grid.

gp_fit utils Fit a squared-exponential Gaussian process and evaluate it on a grid.

gradient utils Derivative $dy/dx$ along the first axis of a sampled profile or map.

make_fit_function utils Build a 1-D parametric model $f(x; c_0, c_1, \dots)$ for profile fitting.

normalize_profile utils Linear normalisation of a profile between its axis and boundary values.

normalized_gradient_scale_length utils Normalised inverse gradient scale length $a/L_y$ of a sampled profile. Convention

trapz_integral utils Definite integral $\int y\,dx$ by the trapezoidal rule.

alpha_heating_power_from_n_D_n_T_T_keV_V equilibrium D-T alpha heating power with the rough $\langle\sigma v\rangle \propto T^2$ fit. Empirical

aspect_ratio_from_a_R equilibrium Aspect ratio $A = R/a = 1/\varepsilon$.

auxiliary_heating_power equilibrium Split auxiliary power into heating and current-drive parts.

ballooning_alpha_from_volume equilibrium Normalised pressure gradient $\alpha$ of local ballooning theory on a general equilibrium. Convention

beta_normal_from_beta_tor equilibrium Normalized beta (Troyon) from toroidal beta, minor radius, field and current.

beta_normal_from_beta_volume equilibrium Normalized volume beta: the volume beta normalized like the Troyon beta_N. Convention

beta_poloidal_from_circumference equilibrium Poloidal beta in the EFIT/OMFIT convention, normalized by the LCFS circumference. Convention

beta_poloidal_from_pressure_integral equilibrium Poloidal beta in the IMAS definition, from the pressure volume integral. Convention

beta_t_from_n_T_B equilibrium Toroidal beta from density, temperature and field. Convention

beta_toroidal_from_p_B0 equilibrium Toroidal beta from the volume-averaged pressure and the vacuum field.

beta_volume_from_p_B2 equilibrium Volume beta: the volume-averaged pressure over the volume-averaged total magnetic energy density. Convention

bootstrap_current_fraction equilibrium Heuristic bootstrap-current fraction $f_{BS}$. Empirical

bremsstrahlung_power_density_from_T_e_p_Z_eff equilibrium Bremsstrahlung power density in the pressure form. Convention

bremsstrahlung_power_density_from_n_e_T_e_Z_eff equilibrium Maxwellian free-free (bremsstrahlung) power density from first principles. Convention

bremsstrahlung_radiation_power_from_z_eff_n_e_t_e equilibrium Bremsstrahlung power density, NRL engineering form. Convention

calc_inverse_aspect_ratio equilibrium Inverse aspect ratio $\varepsilon = a/R_{geo}$ with input validation.

check_kadomtsev_constraint equilibrium Whether engineering exponents satisfy the Kadomtsev constraint within a tolerance.

confinement_factor_ITER89P equilibrium Confinement enhancement factor $H_{89}$ relative to ITER89P.

confinement_scaling_basis equilibrium Energy and power basis a published confinement scaling was fitted on. Convention

confinement_time_from_P_loss_W_th equilibrium Energy confinement time as stored energy over loss power. Convention

confinement_time_from_engineering_parameters equilibrium Thermal energy confinement time from an engineering-parameter scaling law. Empirical Convention

coulomb_logarithm_from_n_T equilibrium Coulomb logarithm $\ln\Lambda$ for electron collisions above 10 eV. Convention

current_density_from_B equilibrium Toroidal current density from the radial derivative of a poloidal field. Convention

current_drive_efficiency equilibrium Heuristic lower-hybrid current-drive efficiency $\eta_{CD}$. Empirical

current_limit_from_beta equilibrium Current figure obtained by substituting $\beta_N$ for $q$ in the cylindrical relation.

current_limit_from_q equilibrium Plasma current at a prescribed edge safety factor, cylindrical approximation.

cyclotron_synchrotron_power_density_scaling_from_n_e_B_t_T_e equilibrium Classical electron-cyclotron emission power density, no reabsorption.

cylindrical_safety_factor_from_R_B_epsilon_I_f_kappa_delta equilibrium Cylindrical safety factor with a shape function. Convention

decay_index_from_bz equilibrium Field decay index $n = -\dfrac{R}{B_Z}\dfrac{\partial B_Z}{\partial R}$. Convention

dimensionless_scaling_coeffs_from_engineering_scaling_coeffs equilibrium Dimensionless scaling indices $(\mu_\rho, \mu_\beta, \mu_\nu)$ from engineering exponents.

eK_from_K equilibrium Elongation parameter $e_K$ of the virial relations.

ec_heating_power_from_I_ec_V_ec equilibrium Electron-cyclotron launched power $P_{ec} = I_{ec}V_{ec}$.

electron_pressure equilibrium Electron thermal pressure from density and temperature, $p_e = n_e T_e$. Convention

elongation_from_RZ_boundary equilibrium Boundary elongation $\kappa$ from the extremal points of a contour. Convention

engineering_exponents_from_dimensionless_coeffs equilibrium Engineering exponents $(\alpha_I, \alpha_B, \alpha_P, \alpha_n, \alpha_R)$ from dimensionless indices.

estimated_q95 equilibrium Estimated $q_{95}$ from global shape, plasma current and toroidal field. Convention

exact_volume_from_RZ_contour equilibrium Plasma volume from a closed $(R, Z)$ contour by Green’s theorem. Convention

flux_perturbation_from_normal_displacement equilibrium Ideal (flux-frozen) perturbed flux of a displacement normal to the flux surfaces. Convention

generalized_parabolic_profile equilibrium Smooth core profile that falls from its axis value to its edge value. Convention

generalized_parabolic_profile_derivative equilibrium Gradient against normalized flux of generalized_parabolic_profile. Convention

generalized_straight_field_line_angle equilibrium Straight-field-line poloidal angle of the generalised family: PEST, Boozer, Hamada, equal-arc. Convention

goldston_l_mode_confinement_time_from_I_P_R_a_kappa equilibrium Goldston L-mode energy confinement time. Empirical Convention

grad_shafranov_source equilibrium Right-hand side of the poloidal-flux equation from Ampere’s law, any region. Convention

heating_power_from_p_ohm_p_aux equilibrium Total heating power $P_{heat} = P_{ohm} + P_{aux}$.

inductive_voltage_from_dW_magdt_I_p equilibrium Inductive voltage from the rate of change of magnetic energy. Convention

internal_inductance_from_W_int_Ip equilibrium Dimensional internal inductance from the poloidal-field energy inside the plasma. Convention

internal_inductance_from_li_3_R0 equilibrium Dimensional internal inductance from the IMAS li_3. Convention

inverse_aspect_ratio_from_a_R equilibrium Inverse aspect ratio $\varepsilon = a/R$.

ion_pressure equilibrium Ion thermal pressure of one species from density and temperature, $p_i = n_i T_i$. Convention

kadomtsev_constraint_from_engineering_exponents equilibrium Residual of the Kadomtsev high-beta constraint on engineering exponents.

kinetic_energy_from_beta_p_B_pa_V_p equilibrium Thermal energy from poloidal beta, $W_K = \tfrac{3}{2}\,\beta_p B_{pa}^2 V_p/(2\mu_0)$. Convention

kink_safety_factor equilibrium Kink safety factor $q_*$ with the Freidberg beta and current limits. Convention

li_1_from_Bp2_volume_integral equilibrium Internal inductance in the circumference-normalized l_i(1) definition.

li_3_from_Bp2_volume_integral equilibrium Internal inductance in the IMAS li_3 definition, from the poloidal-field energy.

li_3_from_internal_inductance_R0 equilibrium IMAS li_3 from a dimensional internal inductance. Convention

line_to_volume_avg_density equilibrium Volume-averaged density from a line average with a fixed profile factor.

loop_voltage_from_total_flux equilibrium Surface loop voltage from the time series of boundary flux. Convention

loss_power_from_p_heat_dWdt_p_rad equilibrium Loss power $P_{loss} = P_{heat} - dW/dt - P_{rad}$. Convention

magnetic_energy_from_li_B_pa_V_p equilibrium Poloidal magnetic energy from the internal inductance, $W_M = l_i B_{pa}^2 V_p/(2\mu_0)$. Convention

miller_surface equilibrium Major radius and height of a Miller-parametrised flux surface. Convention

modified_tanh_profile equilibrium H-mode pedestal profile: Groebner’s modified hyperbolic tangent in normalized flux. Convention

modified_tanh_profile_derivative equilibrium Gradient against normalized flux of modified_tanh_profile. Convention

nbi_heating_power_from_I_nbi_V_nbi equilibrium Neutral-beam injected power $P_{nbi} = I_{nbi}V_{nbi}$.

neo_alcator_confinement_time_from_n_a_R_q equilibrium Neo-Alcator ohmic energy confinement time. Empirical Convention

normalize_psi equilibrium Deprecated: use psi_normalised.

normalized_collisionality_from_a_n_q_epsilon_T equilibrium Collisionality scaling form $\nu_* \propto a\,n\,q/(\varepsilon^{5/2}T^2)$. Convention

normalized_collisionality_from_nu_ii_T_i_M_i_R_a_q equilibrium Ion collisionality $\nu_*$ from a collision frequency. Convention

normalized_larmor_radius_from_M_T_a_Bt equilibrium Normalised ion gyroradius $\rho_* = \rho_i/a$ in SI inputs. Convention

normalized_plasma_current equilibrium Normalised plasma current $I_N = I_p/(a B_t)$ in MA/(m T). Convention

nu_star_from_n_T_B_R_epsilon_kappa_I equilibrium Normalised collisionality $\nu_*$ in the Verdoolaege engineering form. Convention

ohmic_heating_power_from_I_p_V_res equilibrium Ohmic heating power $P_{ohm} = I_pV_{res}$. Convention

ohmic_l_mode_confinement_time_from_tau_ohmic_tau_aux equilibrium Ohmic and auxiliary confinement times combined in quadrature. Convention

omega_i_tau_E_from_B_tau_E_M equilibrium Ion-cyclotron-normalised confinement time $\Omega_i\tau_E$. Convention

peaking_factor equilibrium Profile peaking factor, central value over volume average.

phi_from_Bphi equilibrium Toroidal flux $\Phi$ through a poloidal cross-section. Convention

poloidal_field_factor equilibrium Sauter Eq. 20 prefactor $k = \sigma_{R\varphi Z}\,\sigma_{B_p}/(2\pi)^{e_{B_p}}$. Convention

poloidal_field_magnitude

equilibrium

Poloidal field strength $

B_p

= \sqrt{B_R^2 + B_Z^2}$.

Convention

psi_from_RBtheta equilibrium Poloidal flux $\psi$ from a line integral of $R B_\theta$ across flux surfaces. Convention

psi_normalised equilibrium Normalised poloidal flux $\psi_N$. Convention

q_cyl_from_B_R_epsilon_kappa_I equilibrium Cylindrical safety factor in the Verdoolaege convention. Convention

q_from_flux_surface_averages equilibrium Safety factor $q$ from flux-surface geometric averages and poloidal current. Convention

q_from_phi equilibrium Safety factor $q$ as the flux derivative $d\Phi/d\psi$. Convention

q_from_rhoN equilibrium Safety factor from the toroidal-flux label $\rho_N(\psi_N)$. Convention

q_star_cylindrical equilibrium Menard’s cylindrical safety factor $q^* = \pi a^2 B_0(1+\kappa^2)/(\mu_0 R_0 I_p)$. Convention

q_star_kink equilibrium Freidberg’s kink safety factor $q_* = 2\pi a^2\kappa B_0/(\mu_0 R_0 I_p)$. Convention

r_at_z_extremum_from_RZ_contour equilibrium Major radius where a closed contour reaches its highest or lowest point. Convention

radial_magnetic_field_from_psi equilibrium Radial magnetic field $B_R$ from the poloidal flux map. Convention

rhoN_from_phi equilibrium Normalised toroidal-flux radius $\rho_N$. Convention

rhoN_from_qpsiN equilibrium Normalised toroidal-flux radius from the $q$ profile. Convention

rho_star_from_M_T_B_R_epsilon equilibrium Normalised ion gyroradius $\rho_*$ in the Verdoolaege engineering form. Convention

rho_tor_from_phi

equilibrium

Dimensional toroidal-flux radius $\rho_{tor} = \sqrt{

\Phi

/(\pi

B_0

)}$.

Convention

sfl_toroidal_angle_shift equilibrium Toroidal-angle shift $\nu$ that keeps field lines straight in a non-PEST poloidal angle. Convention

shafranov_shift_from_r_a_R0_beta_p_li equilibrium Large-aspect-ratio Shafranov shift of a circular surface relative to the boundary. Convention

shear_from_r_q equilibrium Magnetic shear $s$ of the safety-factor profile. Convention

shear_from_volume equilibrium Magnetic shear against the volume radius $r_V = \sqrt{V/2\pi^2R_0}$. Convention

spitzer_resistivity_from_T_e_Z_eff_ln_Lambda equilibrium Spitzer parallel resistivity $\eta_\parallel$ (NRL, parallel coefficient). Convention

stored_energy_from_beta_V equilibrium Pressure volume integral $\langle p\rangle V$ from toroidal beta – not the thermal energy. Convention

stored_energy_from_p_V equilibrium Pressure volume integral $\int p\,dV \approx \langle p\rangle V$ – not the thermal (stored) energy. Convention

straight_field_line_angle equilibrium Straight-field-line (PEST) poloidal angle on one flux surface. Convention

surface_poloidal_flux_from_psi_boundary equilibrium Total poloidal flux at the plasma surface, $\Phi_{surface} = 2\pi\psi_b$. Convention

thermal_energy_from_p_V equilibrium Thermal (stored kinetic) energy of the plasma, $W_{th} = \tfrac{3}{2}\int p\,dV$. Convention

toroidal_current_density_from_p_prime_ff_prime equilibrium Plasma toroidal current density allowed by force balance, $J_\phi(p’, FF’)$. Convention

toroidal_electric_field equilibrium Toroidal electric field $E_\varphi = -\dfrac{1}{2\pi R}\dfrac{\partial\psi}{\partial t}$. Convention

toroidal_flux_from_q_psi equilibrium Cumulative toroidal flux $\Phi(\psi)$ from the $q$ profile on a full-weber grid. Convention

triangularity_from_RZ_boundary equilibrium Boundary triangularity $\delta$, the mean of the upper and lower values. Convention

triangularity_lower_from_RZ_boundary equilibrium Lower boundary triangularity $\delta_l$ from the contour’s lowest point. Convention

triangularity_upper_from_RZ_boundary equilibrium Upper boundary triangularity $\delta_u$ from the contour’s highest point. Convention

vacuum_toroidal_field equilibrium Vacuum toroidal field of a set of toroidal-field coils. Convention

verify_kadomtsev_constraint equilibrium Kadomtsev constraint residual of the engineering scaling behind dimensionless indices.

vertical_magnetic_field_from_psi equilibrium Vertical magnetic field $B_Z$ from the poloidal flux map. Convention

volume_from_RZ_boundary equilibrium Plasma volume from a boundary polygon, mean-radius approximation.

approximated_diamagnetism_from_B_pa_B_tv_R0_delta_phi virial Diamagnetic parameter from the vacuum toroidal field and flux change. Convention

virial_D0_boundary_from_bp_li_eK virial Boundary Shafranov shift $D_0(b)$ from $\beta_p$, $l_i$ and elongation. Convention

virial_S1_approx virial Leading-order value of the first Shafranov integral, $S_1 = 2$. Convention

virial_S2_approx_from_D0_a_R0 virial Analytic approximation of the second Shafranov integral for an elongated boundary. Convention

virial_S3_approx_from_eK_d virial Analytic approximation of the third Shafranov integral for an elongated boundary. Convention

virial_alpha_approx_from_kappa virial Virial closure coefficient from the boundary elongation (Bongard $\hat\alpha_1$). Convention

virial_alpha_from_R_Bz_Bp_dl virial Virial closure coefficient from the boundary field (thin-annulus $\hat\alpha_2$). Convention

virial_beta_p_from_S_alpha_mu virial Poloidal beta from the Shafranov integrals, low-aspect-ratio closure. Convention

virial_beta_p_from_S_li virial Poloidal beta from $S_1$, $S_2$ and a known internal inductance.

virial_beta_p_from_volume virial Poloidal beta from a volume integral of pressure. Convention

virial_beta_p_lao_from_S_mu_rt virial Poloidal beta, Lao large-aspect-ratio virial closure. Convention

virial_beta_p_li_from_S_alpha_mu_rt virial Lao closure bundle: $\beta_p$, $l_i$ and $\beta_{p,d}$ in one call.

virial_beta_pd_from_S_mu_rt virial Diamagnetic poloidal beta $\beta_{p,d}$ from $S_1$, $S_2$ and $\mu_i$. Convention

virial_bongard_from_S_alpha_mu virial Low-aspect-ratio virial closure (Bongard 2016): $\beta_p$ and $l_i$.

virial_bp_li_lihat_from_S123 virial Deprecated historical name for the three-relation virial solve.

virial_closure_denominators virial The denominator each virial closure divides by, for conditioning checks.

virial_full_123_from_S_alpha_rt virial Solve all three virial relations for $\beta_p$, $l_i$ and $\mu_i$. Convention

virial_identity_residuals virial Residuals of the three virial identities at a given $(\beta_p, l_i, \mu_i)$. Convention

virial_kinetic_energy virial Bulk kinetic energy of a sampled flow, $W_{kin} = \tfrac{1}{2}\sum n m v^2\,V$.

virial_lao_from_S_alpha_mu_rt virial Large-aspect-ratio virial closure (Lao 1985): $\beta_p$ and $l_i$.

virial_li_from_S_alpha_mu virial Internal inductance from the Shafranov integrals, low-aspect-ratio closure. Convention

virial_li_from_S_alpha_rt virial Internal inductance, Lao large-aspect-ratio virial closure.

virial_li_from_volume virial Internal inductance from a volume integral of $B_p^2$. Convention

virial_magnetic_energy virial Magnetic energy of a sampled field, $W_{mag} = \sum B^2\,V/(2\mu_0)$.

virial_muihat_from_Bt_R0_dphi virial Diamagnetic parameter $\hat\mu_i$ from the measured diamagnetic flux. Convention

virial_normalized_residual virial Scale a virial identity residual so shots can be compared.

virial_pair_12_from_S_mu_rt virial Virial closure on $E_1$ and $E_2$, leaving $E_3$ free. Convention

virial_pair_13_from_S_alpha_mu virial Virial closure on $E_1$ and $E_3$, leaving $E_2$ free. Convention

virial_pair_23_from_S_alpha_mu_rt virial Virial closure on $E_2$ and $E_3$, leaving $E_1$ free.

virial_residual_rms virial Aggregate the three normalised virial residuals into one number.

virial_stability_criterion virial Deprecated: the virial ratio minus an unsourced threshold 0.5; use virial_theorem.

virial_theorem virial Total energy and virial ratio of magnetic, kinetic and thermal contributions.

virial_thermal_energy virial Thermal energy of a sampled plasma, $W_{th} = \tfrac{3}{2}\sum n T\,V$. Convention

ballooning_alpha_from_p_B_R stability Normalised pressure gradient $\alpha$ of the $s$-$\alpha$ ballooning model. Convention

ballooning_radial_wavenumber stability Radial wavenumber of a ballooning mode along the field line, in the $s$-$\alpha$ local frame. Convention

ballooning_stability_criterion stability Distance from the first ballooning stability boundary $\alpha_{crit} \approx 0.6\,s$. Empirical

beta_N_from_beta_a_B0_Ip stability Normalised beta in the Troyon convention, %·m·T/MA. Convention

beta_pol_from_beta_tor stability Poloidal beta from toroidal beta, cylindrical relation.

beta_stability_boundary stability Deprecated: fraction-convention twin of kink_stability_criterion; use the troyon boundary. Empirical

beta_tor_from_beta_pol stability Toroidal beta from poloidal beta, cylindrical relation.

bussac_internal_kink_energy stability Reduced toroidal $m = n = 1$ internal-kink energy of Bussac et al., up to its positive prefactor. Convention

bussac_poloidal_beta stability Bussac’s poloidal beta inside the $q = 1$ surface. Convention

c_s_from_Te_Ti_mi stability Isothermal ion sound speed $c_s = \sqrt{(T_e + T_i)/m_i}$. Convention

collisionality_from_n_T_B_R stability Electron collisionality figure $6.921\times10^{-18}\,n_e R_0/(T_e^2 B_t)$. Convention

delta_prime_from_outer_derivatives stability Tearing stability index from the outer solution’s derivatives at the rational surface. Convention

empirical_li_qa stability Vertices of the JET empirical $l_i$-$q_\psi$ stability boundary (Wesson 1989, Fig. 6). Empirical

field_line_label stability Clebsch field-line label on a flux surface in straight-field-line coordinates. Convention

ggj_ideal_interchange_index stability Glasser–Greene–Johnson ideal-interchange (Mercier) index $D_I$ from the layer coefficients. Convention

ggj_resistive_interchange_index stability Glasser–Greene–Johnson resistive-interchange index $D_R$ from the layer coefficients. Convention

ggj_resistive_interchange_index_from_ideal stability GGJ resistive-interchange index $D_R$ from the ideal index $D_I$ and $H$. Convention

greenwald_density stability Greenwald density limit $n_G$. Empirical Convention

greenwald_fraction stability Greenwald fraction $f_G = n_e/n_G$. Convention

helical_harmonic stability The real perturbation carried by one complex $m/n$ Fourier coefficient. Convention

helical_phase stability Helical phase of an $m/n$ perturbation at a point on a flux surface. Convention

island_pendulum_hamiltonian stability Local flux function of a constant-$\psi$ magnetic island. Convention

island_separatrix_half_width stability Radial half-width of the island separatrix at a given helical phase. Convention

island_width_from_resonant_flux stability Full magnetic island width from the pitch-resonant flux. Convention

kadomtsev_mixing_radius stability Kadomtsev mixing radius: where the $m/n = 1/1$ helical flux returns to its axis value. Convention

kink_stability_criterion stability Deprecated: heuristic margin against an unsourced $\beta_{N,crit} = 2.8\,q_{95}$; use the troyon boundary. Empirical

li_from_qa_empirical stability Internal inductance at a given $q_a$ by interpolating the JET survey points. Empirical

magnetic_well_from_specific_volume stability Normalised magnetic well from the specific volume $V’$ against toroidal flux. Convention

magnetosonic_phase_speeds stability Fast and slow magnetosonic phase speeds at angle $\theta$ to $\mathbf B_0$.

mercier_criterion_circular stability Mercier’s interchange criterion for a large-aspect-ratio circular tokamak. Convention

plasma_stability_margins stability Deprecated: beta, $q_{95}$ and density margins built on the unsourced $0.028\,q_{95}$ beta limit. Convention

power_limit_from_beta stability Energy-like figure $\beta_N B_0^2 V/(2\mu_0)$ labelled a power limit.

power_limit_from_q stability Expression $2\pi R_0 I_p/(\mu_0 q_{95})$ labelled a power limit.

resonant_flux_from_delta stability Pitch-resonant flux from the resonance parameter and the surface geometry. Convention

rhostar_from_Te_a_Bt stability Normalised electron gyroradius $\rho_e/a$. Convention

s_alpha_ballooning_eigenmode stability The most unstable localised eigenmode of the $s$-$\alpha$ ballooning equation with inertia. Convention

s_alpha_ballooning_solution stability The even solution $F(\theta)$ of the marginal $s$-$\alpha$ ballooning equation. Convention

s_alpha_ballooning_stable stability Ideal ballooning stability of the circular $s$-$\alpha$ model. Convention

s_alpha_curvature_drive stability Normal-curvature drive of the $s$-$\alpha$ ballooning equation along the extended angle. Convention

s_alpha_marginal_alpha stability First and second ballooning stability boundaries of the $s$-$\alpha$ model. Convention

sawtooth_stability_criterion stability Deprecated: heuristic margin against an unsourced $\beta_{p,crit} = 0.3\,(1 - q_0)$; no replacement. Empirical

shear_alfven_frequency stability Frequency of the ideal shear Alfven wave in a uniform plasma.

slab_perturbed_flux stability Helical flux of a sheared slab with a tearing- or twisting-parity perturbation. Convention

suydam_criterion stability Suydam’s local interchange criterion of a cylindrical screw pinch. Convention

v_alfven_from_B_n_mi stability Alfven speed $v_A = B/\sqrt{\mu_0 n m_i}$.

calculate_distance green Euclidean distance between $(r_1, z_1)$ and $(r_2, z_2)$ in the poloidal plane.

complete_elliptic_integral_e green Complete elliptic integral of the second kind $E(m)$. Convention

complete_elliptic_integral_k green Complete elliptic integral of the first kind $K(m)$. Convention

elliptic_integral green Polynomial approximations of $K$ and $E$ for a ring source and observer points.

green_br_bz green Field of a unit-current ring, $(B_r, B_z)$ at observer points (legacy elliptic path). Convention

green_br_bz_exact green $(B_r, B_z)$ per unit ring current, exact elliptic integrals, broadcasting. Convention

green_psi_exact green Poloidal flux per unit ring current, exact elliptic integrals. Convention

green_r green Poloidal flux per unit ring current at observer points (legacy elliptic path). Convention

greens_function_2d green Axisymmetric kernel $\sqrt{RR_0}\,K(m)$ between a field point and a ring source. Convention

greens_function_3d green Toroidal-angle-resolved variant of the $\sqrt{RR_0}\,K(m)$ kernel.

greens_function_exact green Exact free-space axisymmetric Green’s function and its derivatives. Convention

greens_integral_2d green Line integral along $R$ of the 2-D kernel times a source density.

greens_integral_3d green Line integral along $R$ of the 3-D kernel times a source density.

mutual_inductance green Mutual inductance between two rectangular-cross-section ring coils.

self_inductance green Self-inductance of a rectangular-cross-section ring coil.

trapz_integral green Trapezoidal integral $\int y\,dx$, local copy for the Green kernels.

coronal_relaxation_time atomic Slowest relaxation time of the ionisation balance towards coronal equilibrium.

fractional_abundances atomic Ionisation-equilibrium (coronal) fractional abundances from ACD and SCD tables. Convention

hydrogenic_energy_level atomic Bohr energy of level $n$ of a one-electron ion, with the reduced-mass correction.

hydrogenic_transition_wavelength atomic Vacuum wavelength of the $n_u \to n_l$ line of a one-electron ion (Rydberg formula). Convention

impurity_fraction_from_effective_charge atomic Impurity-to-electron density ratio that produces a given effective charge.

interpolate_adf11 atomic Interpolate every charge-state block of an ADF11 table at given $(n_e, T_e)$. Convention

line_cooling_coefficient atomic Equilibrium line-radiation cooling coefficient $L_{line}(n_e, T_e)$ of a species. Convention

mean_charge_from_charge_state_densities atomic Mean charge of one element over its charge states. Convention

mean_square_charge_from_charge_state_densities atomic Mean square charge of one element over its charge states. Convention

transient_fractional_abundances atomic Fractional abundances a time tau after neutral atoms enter a plasma of fixed n_e, T_e.

z_eff_from_n_s_Z_s atomic Effective charge of a mixture of ion species. Convention

bias_standard_error statistics Standard error of the mean of $n$ unit-variance samples, $1/\sqrt{n}$.

chi_squared statistics Chi-square, $\sum_i (r_i/\sigma_i)^2$ over the finite terms.

dynamic_range statistics Sample range, $\max - \min$ over the finite entries.

fractional_rms_improvement statistics Fractional RMS improvement, $1 - \mathrm{RMS}(r)/\mathrm{RMS}(r_0)$.

lag1_autocorrelation statistics Lag-1 autocorrelation of the finite entries, in the order given.

linear_trend statistics Least-squares slope of $y$ against $x$, in units of $y$ per unit $x$.

log10_decay_rate statistics Least-squares slope of $\log_{10}x$ against iteration index over the tail.

median_absolute_deviation

statistics

Median absolute deviation, $\mathrm{median}(

x - \mathrm{median}(x)

)$.

monotonic_fraction statistics Fraction of consecutive steps in which the sequence decreased.

noise_band statistics Mean and population standard deviation of a reference (effect-free) sample.

normalized_residual statistics Residuals in units of the fitted uncertainty, $z = r\,w/k$.

outlier_fraction

statistics

Fraction of finite entries with $

x_i

> \mathrm{level}$.

pearson_correlation statistics Pearson correlation of two series over their pairwise-finite samples.

percentile_scale statistics A high percentile of |x| over the finite entries, an amplitude scale one anomalous sample cannot set.

reduced_chi_squared statistics Reduced chi-square, $\chi^2/\nu$ per degree of freedom.

relative_spread

statistics

Relative spread, $(\max - \min)/\max

$ over the finite entries.

residual_bias statistics Mean of the finite entries, the systematic offset of a residual sample.

rms statistics Root mean square, $\sqrt{\mathrm{mean}(x_i^2)}$, over the finite entries.

robust_z_scores statistics Deviation from the median in robust sigma units, sample by sample.

runs_test_z statistics Wald-Wolfowitz runs-test z-score for the sign sequence of a sample.

sigma_threshold_crossing statistics First time outside the reference window where the signal leaves its noise band.

sigma_unit_factor statistics Recover the units-of-fit factor $k$ from a fit’s own per-channel chi-square.

probe_axis magnetics Direction cosines (c_R, c_Z) of a poloidal-field probe’s sensitive axis.

project_poloidal_field magnetics The component of (B_R, B_Z) a probe oriented at poloidal_angle reads.

banana_width neoclassical Radial width of a thermal banana orbit, the conventional estimate. Convention

collisions_per_transit neoclassical Collision frequency in units of the transit frequency, $\hat\nu = qR_0\nu/v$. Convention

coulomb_logarithm_electron_sauter neoclassical Electron Coulomb logarithm $\ln\Lambda_e$ in the Sauter convention. Convention

coulomb_logarithm_ion_sauter neoclassical Ion-ion Coulomb logarithm $\ln\Lambda_{ii}$ in the Sauter convention. Convention

deeply_trapped_bounce_frequency neoclassical Bounce frequency of a deeply trapped particle, the small-amplitude limit of the banana orbit. Convention

electron_collisionality_sauter neoclassical Normalised electron collisionality $\nu_e^*$ in the Sauter convention. Convention

ion_collisionality_sauter neoclassical Normalised ion collisionality $\nu_i^*$ in the Sauter convention. Convention

neoclassical_regime_boundaries neoclassical Banana–plateau and plateau–Pfirsch–Schl"uter orderings in $\hat\nu$. Convention

redl_bootstrap_coefficients neoclassical The Redl 2021 bootstrap coefficients $L_{31}$, $L_{32}$, $L_{34}$, $\alpha$. Empirical Convention

redl_bootstrap_current neoclassical Flux-surface-averaged bootstrap current $\langle j_\parallel B\rangle$, Redl 2021. Empirical Convention

redl_neoclassical_conductivity neoclassical Neoclassical parallel conductivity $\sigma_{\mathrm{neo}}$, Redl 2021. Empirical Convention

sauter_bootstrap_coefficients neoclassical The Sauter 1999 bootstrap coefficients $L_{31}$, $L_{32}$, $L_{34}$, $\alpha$. Empirical Convention

sauter_bootstrap_current neoclassical Flux-surface-averaged bootstrap current $\langle j_\parallel B\rangle$, Sauter 1999. Empirical Convention

sauter_neoclassical_conductivity neoclassical Neoclassical parallel conductivity $\sigma_{\mathrm{neo}}$, Sauter 1999. Empirical Convention

sauter_spitzer_conductivity neoclassical Spitzer parallel conductivity $\sigma_{\mathrm{Sptz}}$ in the Sauter normalisation. Empirical Convention

transit_frequency neoclassical Poloidal transit frequency of a passing particle on a large-aspect-ratio circular surface. Convention

trapped_particle_effective_collision_frequency neoclassical Rate at which pitch-angle scattering detraps a trapped particle. Convention

trapped_particle_fraction neoclassical Trapped-particle fraction $f_t$ of a circular surface of inverse aspect ratio. Empirical Convention

atomic_inventory_from_molecular_gas startup Atom inventory of a molecular gas fill. Convention

boundary_loop_voltage_terms_from_L_e_I_p_M_pj_I_j startup Boundary loop voltage split into the four ways the boundary flux can change. Convention

breakdown_margin startup Available drive relative to the Lloyd breakdown threshold. Convention

critical_ionization_fraction_from_V_p_V_V startup Ionisation fraction at the top of the radiation-ionisation barrier. Convention

d_plasma_inductance_dR_hirshman_from_R_eps_kappa_li startup Radial derivative of the Hirshman plasma inductance. Convention

ejiri_f3_from_alpha startup Low-temperature geometry factor of the Ejiri mirror proxy. Convention

ejiri_mirror_alpha_from_R_S_R_LIN_R_C_Z_max startup Slope of the Ejiri confined-orbit boundary in velocity space. Convention

electron_cyclotron_resonance_field startup Field magnitude at which electrons gyrate in step with a microwave source.

electron_cyclotron_resonance_radius startup Major radius of the electron-cyclotron resonance in a vacuum toroidal field. Convention

flux_closure_margin_from_E_t_a_B_stray_eta startup Order-of-magnitude margin for a current channel to overcome stray field. Convention

internal_inductive_voltage_terms_from_L_i_I_p startup Inductive voltage of the internal inductance, split into ramp and profile terms. Convention

ionization_fraction_from_n_e_n_D0 startup Ionisation fraction of a partially ionised hydrogenic plasma. Convention

lloyd_breakdown_field startup Lloyd threshold toroidal field for hydrogenic tokamak breakdown. Empirical Convention

lloyd_figure_of_merit startup Lloyd’s breakdown figure of merit: toroidal drive times field-line pitch. Convention

lr_time_from_L_R startup Characteristic time of an inductive-resistive circuit. Convention

neutral_density_after_ionization_from_n_0_n_e_V_p_V_V startup Neutral atom density left in the vessel after some has been ionised. Convention

neutral_density_from_pressure startup Number density of a fill gas at a measured pressure and temperature. Convention

plasma_current_derivative_lumped_from_V_loop_R_p_I_p_L_p startup Rate of change of plasma current in a lumped single-loop circuit. Convention

plasma_external_inductance_hirshman_from_R_eps_kappa startup External inductance of a shaped plasma, Hirshman and Neilson’s fit. Convention

plasma_inductance_circular_from_R0_a_li startup Self-inductance of a high-aspect-ratio plasma ring, Mitarai’s form. Convention

plasma_inductance_hirshman_from_R_eps_kappa_li startup Total plasma inductance, Hirshman external plus the internal term. Convention

plasma_resistance_uniform_ellipse_from_eta_R0_a_kappa startup Loop resistance of a uniform-current elliptical plasma ring. Convention

plasma_self_field_from_I_p_a startup Poloidal field a circular current channel makes at its own edge. Convention

radiation_ionization_barrier_from_P_RI_n_0_V_V startup Height of the radiation-ionisation barrier a burn-through must cross. Convention

radiation_ionization_power_from_P_RI_n_e_n_D0_V_p startup Radiation and ionisation power lost by a partially ionised plasma. Convention

resistivity_from_n_e_nu_e startup Resistivity from the electron momentum-transfer collision frequency. Convention

startup_geometry_from_limiter_radii startup Geometric centre and minor radius of the limiter aperture. Convention

townsend_breakdown_field startup Breakdown threshold field of a gas, by inverting the Townsend closure. Convention

townsend_breakdown_field_for_gas startup Breakdown threshold field of a catalogued fill gas, pressure in pascal. Empirical Convention

townsend_coefficients_for_gas startup Townsend similarity coefficients of a catalogued fill gas, in SI. Empirical Convention

townsend_ionization_coefficient startup Townsend first ionisation coefficient of a gas in a field. Convention

vertical_field_from_I_p_R0_a_beta_p_li startup Vertical field that holds a circular plasma in radial equilibrium. Convention

current_weighted_flux_from_psi_j_dS transformer Current-weighted average of the poloidal flux over the plasma cross-section. Convention

equilibrium_surface_voltage_from_I_p_dL_i_V_R transformer Voltage at the equilibrium flux surface, inferred from the inductance history. Convention

equilibrium_surface_voltage_from_L_i_dI_p_V_B_V_R transformer Voltage at the equilibrium flux surface, inferred from the current history. Convention

internal_inductance_from_psi_C_psi_B_I_p transformer Dimensional internal inductance from the equilibrium and boundary fluxes. Convention

internal_inductance_rate_from_I_p_V_R_V_C transformer Rate of change of the internal inductance, Romero’s exact balance. Convention

plasma_current_rate_from_L_i_V_B_V_C_V_R transformer Rate of change of the plasma current, Romero’s exact balance. Convention

resistive_voltage_from_R_p_I_p_I_ni transformer Resistive voltage of the plasma, net of non-inductive current drive. Convention

romero_closure_rates_from_I_p_L_i_V_CB_V_B_V_R_k_tau transformer State rates of Romero’s first-order current-diffusion closure. Empirical Convention

boris_orbit particle Charged-particle orbit from the Lorentz force, by the Boris scheme. Convention

bounce_harmonic_detuning particle Frequency mismatch of a trapped orbit with a toroidal-$n$ perturbation at bounce harmonic $\ell$. Convention

canonical_toroidal_momentum particle Canonical toroidal angular momentum of a particle in an axisymmetric field. Convention

curvature_drift_velocity particle Curvature drift velocity of the guiding centre. Convention

exb_drift_velocity particle E-cross-B drift velocity of the guiding centre. Convention

grad_b_drift_velocity particle Grad-B drift velocity of the guiding centre. Convention

guiding_center_toroidal_momentum particle Canonical toroidal momentum of a guiding centre, flux plus mechanical parts. Convention

gyration_offset particle Position of a gyrating particle relative to its guiding centre. Convention

gyrofrequency particle Signed gyrofrequency of a charged particle. Convention

larmor_radius particle Larmor (gyro) radius of a charged particle. Convention

magnetic_moment particle Magnetic moment of a gyrating particle, the first adiabatic invariant. Convention

parallel_speed_from_mu particle Parallel speed where the field is $B$, from energy and magnetic-moment conservation. Convention

psi_per_radian_from_cocos particle The guiding-centre $\psi = RA_\phi$ from a flux stored in a COCOS convention. Convention

cylindrical_current_diffusion_rate geometry Rate of change of the enclosed current under resistive diffusion of the poloidal field. Convention

cylindrical_enclosed_current geometry Axial current inside each radius of a straight current column. Convention

cylindrical_internal_inductance geometry Normalized internal inductance of a current column of radius $a$. Convention

cylindrical_parallel_wavenumber geometry Parallel wavenumber of an $m/n$ harmonic on a cylindrical surface of safety factor $q$. Convention

cylindrical_poloidal_field geometry Poloidal field of a straight current column by Ampère’s law. Convention

cylindrical_poloidal_flux geometry Poloidal flux per radian of a straightened torus, zero on the axis. Convention

cylindrical_safety_factor_from_r_B geometry Safety factor of a straight cylinder standing in for a torus. Convention

harris_sheet_current_density geometry Current density of a Harris current sheet, from Ampere’s law. Convention

harris_sheet_field geometry Reconnecting field of a Harris current sheet. Convention

local_slab_from_cylinder geometry The field-aligned sheared slab of a cylinder at $r_0$: $(k_y, L_s, k_z)$ of an $m/n$ harmonic. Convention

peaked_current_safety_factor geometry Cylindrical $q$ profile of the peaked current $j \propto (1 - x^2)^\nu$. Convention

shear_length_from_q_R0_s geometry Magnitude of the shear length a torus or cylinder gives its local slab. Convention

sheared_slab_field geometry Magnetic field of the sheared slab at radial position $x$. Convention

sheared_slab_parallel_wavenumber geometry Parallel wavenumber across a sheared slab, to first order in $x/L_s$. Convention

slab_parallel_wavenumber geometry Component of a wavevector along the magnetic field. Convention

x_point_flux geometry Flux function of a current-free magnetic X-point. Convention

gwb_stochastic_threshold ripple Goldston–White–Boozer ripple above which banana-tip motion is stochastic. Convention

gwb_stochasticity_parameter ripple Ripple over its GWB threshold: stochastic banana tips where it exceeds 1. Convention

ripple_amplitude ripple Ripple amplitude from the extreme field strengths along the toroidal angle. Convention

ripple_trapping_pitch

ripple

Largest pitch $

v_\parallel/v

$ trapped in a ripple well of depth $\delta$.

Convention

ripple_well_parameter ripple Whether ripple makes local wells along a field line: $\alpha^* < 1$. Convention

toroidal_ripple_field ripple Field strength on a flux surface with toroidicity and TF-coil ripple. Convention

avalanche_efolds_from_current_drop disruption Number of avalanche e-folds a current quench provides, $E \gg E_c$.

avalanche_growth_rate disruption Rosenbluth–Putvinski avalanche (secondary) growth rate of the runaway density. Convention

connor_hastie_critical_field disruption Connor–Hastie critical field below which no electron runs away.

current_quench_current disruption Exponential (L/R) current quench of the plasma current.

dreicer_field disruption Dreicer field: the drag on a thermal electron. Convention

dreicer_generation_rate disruption Dreicer (primary) runaway generation rate, Connor–Hastie asymptotic form. Convention

inductive_parallel_electric_field disruption Parallel electric field a changing plasma current induces in the plasma. Convention

relativistic_collision_time disruption Collision time of a relativistic electron, $\tau_c = m_ec/(eE_c)$.

runaway_critical_momentum disruption Critical momentum above which an electron runs away, in units of $m_ec$.

runaway_current_from_density disruption Current carried by runaway electrons moving at the speed of light.

thermal_quench_temperature disruption Prescribed exponential thermal quench of the electron temperature.

beam_birth_probability_density nbi Fast-ion birth probability per unit path length along the beam. Convention

beam_particle_rate_from_power_energy nbi Injected particle rate of one monoenergetic beam component. Convention

injected_toroidal_angular_momentum_rate nbi Toroidal angular momentum carried into the machine per second by the injected particles. Convention

neutral_beam_optical_depth nbi Optical depth accumulated along the beam path, by the trapezoidal rule. Convention

neutral_survival_fraction_from_optical_depth nbi Fraction of the injected neutrals not yet ionised.

shine_through_fraction nbi Fraction of the injected neutrals that leave the path un-ionised. Convention

cma_coordinates waves CMA-diagram coordinates of a cold electron plasma. Convention

cold_plasma_refractive_index_squared waves The two roots $n^2$ of the cold-plasma dispersion relation at angle $\theta$ to $\mathbf B_0$. Convention

dielectric_tensor waves Cold-plasma dielectric tensor in the frame with $\mathbf B_0 \parallel \hat z$. Convention

perpendicular_refractive_index_squared waves Ordinary and extraordinary modes at perpendicular propagation. Convention

plasma_frequency waves Plasma frequency of one species. Convention

propagation_regime waves Classify a refractive index squared: propagating, evanescent, cutoff or resonance. Convention

stix_parameters waves Stix parameters of a cold, multi-species, magnetised plasma. Convention

nonambipolar_torque_density ntv Flux-surface-averaged toroidal torque density on the plasma from non-ambipolar radial fluxes (NTV). Convention

ntv_precession_frequency ntv Bounce-averaged toroidal precession of trapped particles, whose zero is the superbanana-plateau resonance. Convention

halo_current_fraction vde Halo-current fraction: peak poloidal halo current over the pre-disruption plasma current. Convention

thin_wall_time vde Resistive time of a thin cylindrical wall, $\tau_w = \mu_0\sigma db$. Convention

toroidal_peaking_factor vde Toroidal peaking factor of the halo current: maximum over the toroidal mean. Convention

vde_growth_rate

vde

Local growth rate of a vertical displacement, $d\ln

Z - Z_0

/dt$.

Convention

vertical_velocity vde Vertical velocity of the current centroid, $dZ/dt$.

wall_mode_decay_time vde Decay time of a thin wall’s poloidal-harmonic-$m$ eddy current.

binary_collision_energy_transfer_factor pwi Largest fraction of a projectile’s energy one elastic collision can give a target atom.

mean_reflected_energy_fraction pwi Mean energy of a reflected particle, as a fraction of the incident energy. Convention

recycling_coefficient pwi Recycling coefficient: particles returned to the plasma per incident particle. Convention

sputtering_threshold_bohdansky pwi Physical-sputtering threshold energy, Bohdansky’s empirical fit. Convention

blob_collisionality sol The collisionality that decides whether a filament stays connected to the sheaths. Convention

blob_crossover_size sol Normalized size at which the inertial and sheath-connected limits give the same velocity. Convention

blob_density_perturbation sol The prescribed filament state: a Gaussian density perturbation on a background. Convention

blob_reference_size sol The blob size at which sheath and polarization closure balance. Convention

blob_reference_velocity sol The blob velocity scale at the reference size. Convention

blob_regime_velocities sol The normalized velocity of each regime of the two-region (midplane plus X-point) model. Convention

eich_integral_width sol Integral power width of the Eich profile, mapped to the outer midplane. Empirical Convention

eich_target_heat_flux_profile sol Target heat-flux profile: an exponential SOL decay convolved with Gaussian spreading. Convention

inertial_blob_velocity sol Radial velocity of a blob whose polarization current closes across the field (inertia). Convention

interpolated_blob_velocity sol Normalized blob velocity bridging the inertial and sheath-connected limits. Convention

ion_saturation_current_density sol Ion saturation current density collected by a negatively biased surface. Convention

ion_sound_speed sol Ion sound speed with the charge and the ion closure explicit. Convention

radiative_condensation_growth_rate sol Growth rate of the thermal-condensation (MARFE) instability at constant pressure. Convention

radiative_thermal_instability_growth_rate sol Growth rate of the radiative thermal instability of a flute perturbation at constant density. Convention

sheath_connected_blob_velocity sol Radial velocity of a blob whose polarization current closes through the sheaths. Convention

sheath_heat_flux sol Parallel heat-flux density into the target sheath. Convention

sheath_particle_flux sol Particle flux density into the target sheath. Convention

spitzer_harm_parallel_heat_flux sol Classical parallel electron heat conduction. Convention

two_point_upstream_temperature sol Upstream temperature from conduction at constant parallel heat flux. Convention

boundary_curve boundaries A boundary sampled along one input, as $(x, y) = (x_\mathrm{sweep}, b)$ arrays. Convention

boundary_value boundaries Value of a boundary’s target quantity at the given inputs. Convention

cylindrical_kink_coordinates boundaries Menard’s cylindrical safety factor, $q^* = \pi a^2 B_{T0}(1+\kappa^2)/(\mu_0 R_0 I_P)$. Convention

evaluate_boundary boundaries Distance and side of an operating value relative to a boundary. Convention

evaluate_window boundaries Position of an operating value relative to an access window. Convention

get_boundary boundaries A registered boundary or window by key.

hugill_coordinates boundaries Hugill-diagram coordinates of an operating state: $(\bar n_e R/B_T,\ 1/q_\mathrm{cyl})$. Convention

hugill_coordinates_st boundaries Spherical-tokamak Hugill coordinates $(\bar n_e R/B_T,\ 1/q_\mathrm{cyl}^{ST})$ of Sykes et al. (2000). Convention

iter_q95_coordinates boundaries The ITER design-guideline $q_{95}$ estimate from global shape, $q_{95} \approx q_* f(\epsilon)$. Convention

kink_coordinates boundaries Freidberg’s kink safety factor of an elongated tokamak, $q_* = 2\pi a^2\kappa B_0/(\mu_0 R_0 I)$. Convention

list_boundaries boundaries Keys of the registered boundaries, optionally restricted to one family.

propagate_boundary_uncertainty boundaries Propagate measured-input uncertainty into a boundary value and a state’s margin (#1874).

same_quantity boundaries Whether two quantities are the same physical quantity in the same unit. Convention

start_q95_coordinates boundaries The START low-aspect-ratio $q_{95}$ estimate of Akers et al. (2000) from global shape. Convention

threshold_curve boundaries A threshold boundary drawn as a straight line across a 2-D projection. Convention

dilution_fraction_from_species impurity Main-ion dilution from an explicit impurity species list. Convention

dilution_fraction_from_zeff impurity Main-ion dilution from Z_eff and one effective impurity charge.

expand_effective_impurity impurity Species densities of a known relative composition behind a pseudo-impurity.

impurity_mixture_moments impurity Charge and mass moments of a normalised impurity mixture. Convention

main_ion_density_from_species impurity Main-ion density that keeps a plasma with these impurities quasi-neutral.

main_ion_density_from_zeff impurity Main-ion density from n_e, Z_eff and one effective impurity charge.

main_ion_fraction_from_zeff impurity Main-ion fraction of the electron density from Z_eff and one effective impurity charge.

reduce_impurity_mixture impurity The single pseudo-impurity that keeps a mixture’s charge, Z^2 and mass moments.

solve_impurity_mixture_for_target_zeff impurity Impurity and main-ion density fractions that give a target effective charge.

dimension_matrix dimensional Dimension matrix $D$ of a list of variables, exactly. Convention

dimensionless_groups dimensional A basis of the Buckingham-Pi groups of a set of variables: $\ker D$. Convention

express_in_basis dimensional Exact coefficients $\mathbf c$ with $\mathbf t = \sum_k c_k \mathbf b_k$. Convention

monomial_dimension dimensional Base dimensions of the monomial $\prod_j x_j^{a_j}$: $D\mathbf a$. Convention

rational_null_space dimensional Exact basis of the null space ${\mathbf a : M\mathbf a = 0}$ of a rational matrix. Convention

similarity_constraint_vectors dimensional Linear constraints a law depending only on kept_groups imposes on its exponents. Convention

state_exponents_from_engineering_exponents dimensional Exponents of $\tau_E$ over $(n, T, B, R)$ implied by an engineering scaling, exactly. Convention

finite_difference_jacobian sensitivity Jacobian of a vector function by forward or central differences.

linear_covariance_propagation sensitivity Output covariance of a linearized model, correlations included.

linearity_ratio sensitivity How much of a finite response the linearization misses.

monte_carlo_propagation sensitivity Output mean and covariance by sampling Gaussian inputs through the full model.

propagate_formula_uncertainty sensitivity Propagate input uncertainty through one vaft.formula function, correlations included.

singular_value_spectrum sensitivity Singular values, numerical rank and null space of a (column-scaled) Jacobian.

alfven_time ordering Alfven transit time across a characteristic length. Convention

braginskii_electron_collision_time ordering Braginskii electron collision time. Convention

braginskii_ion_collision_time ordering Braginskii ion collision time. Convention

debye_length ordering Electron Debye length. Convention

evolution_time ordering Evolution time of a state variable from its rate of change. Convention

inertial_length ordering Inertial (skin) length of a species: the light speed over its plasma frequency. Convention

knudsen_number ordering Knudsen number: mean free path over a characteristic length. Convention

lundquist_number ordering Lundquist number: resistive diffusion time over Alfven time. Convention

mach_number ordering Mach number of a flow against a reference speed. Convention

magnetic_reynolds_number ordering Magnetic Reynolds number: flux advection against resistive diffusion. Convention

magnetization ordering Magnetization of a species: gyro-orbits per collision time. Convention

mean_free_path ordering Mean free path: thermal speed times collision time. Convention

pressure_anisotropy ordering Pressure anisotropy relative to the scalar pressure. Convention

resistive_diffusion_time ordering Resistive diffusion time of magnetic field across a characteristic length. Convention

sound_gyroradius ordering Ion-sound gyroradius: the ion sound speed over the ion gyrofrequency. Convention

thermal_speed ordering Thermal speed of a species. Convention

critical_energy_from_T_e_A_b_n_species fast_ion Critical energy of a fast ion of mass number A_b: E_c = m_b v_c^2 / 2.

critical_velocity_from_T_e_n_species fast_ion Critical speed at which a fast ion’s drag on electrons equals its drag on the field ions.

fast_ion_density_from_source fast_ion Fast-ion density held by a steady source slowing down classically.

fast_ion_energy_density_from_source fast_ion Kinetic-energy density held by a steady source slowing down classically.

fast_ion_pressure_from_energy_density fast_ion Scalar fast-ion pressure of an isotropic fast population: p_f = 2 W_f / 3. Convention

slowing_down_distribution fast_ion Steady isotropic classical slowing-down distribution of a monoenergetic source.

slowing_down_time_between_speeds fast_ion Time a fast ion takes to slow from one speed to a lower one under classical drag.

slowing_down_time_from_T_e_n_e_A_b_Z_b fast_ion Spitzer slowing-down time of a fast ion on electrons. Convention

braginskii_gamma1_perp_from_Z kinetic Braginskii’s electron perpendicular heat-conductivity coefficient $\gamma_1’(Z)$. Convention

classical_electron_heat_diffusivity_from_T_e_B kinetic Braginskii perpendicular electron heat diffusivity, strongly magnetised limit. Convention

classical_ion_heat_diffusivity_from_T_i_B kinetic Braginskii perpendicular ion heat diffusivity, strongly magnetised limit. Convention

electron_collision_time_from_T_e_n_species kinetic Braginskii electron collision time against a mixture of ion species. Convention

electron_ion_energy_exchange_time_from_T_e_n_species kinetic Time in which electrons and a mixture of ion species exchange thermal energy.

ion_collision_time_from_T_i_n_species kinetic Braginskii collision time of one ion species against a mixture of field ions. Convention

ion_ion_coulomb_logarithm_from_n_T kinetic Coulomb logarithm for collisions between ions of one species (NRL). Convention

predator_prey_fixed_point turbulence Positive (coexistence) fixed point of the predator–prey model. Convention

predator_prey_invariant turbulence Conserved quantity of the classical predator–prey model. Convention

predator_prey_period turbulence Small-amplitude oscillation period about the positive fixed point. Convention

predator_prey_response_lag turbulence Small-amplitude delay of the zonal-flow peak after the turbulence peak. Convention

predator_prey_rhs turbulence Time derivatives of the turbulence–zonal-flow predator–prey model. Convention

evaluate_regularized_profile regularized_profiles Evaluate a positive regular flux profile or its flux derivative.

fit_regularized_profile regularized_profiles Fit independent samples with Huber residuals and quadratic curvature.

profile_basis regularized_profiles Evaluate the positive profile basis or its flux-coordinate derivatives.

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From a checkout of the develop branch, run:

python -m vaft.formula.catalog --output docs/_data/formula_catalog.yml

The snapshot records the SHA-256 of every vaft/formula/*.py source file; documentation validation compares them when VAFT_REGISTRY_SOURCE points to the corresponding source checkout. The same text is available offline as vaft.formula.describe("<name>"), vaft.formula.search("<text>") and vaft.formula.list_formulas(category="<category>").

What these objects mean scientifically – which concept a plot draws, which diagnostic measures it, which Data Dictionary path represents it – is generated in the scientific ontology explorer.

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