Characteristic scales and orderings

This page answers one question: how does a plasma state become a statement about which model may describe it? VAFT spreads the answer over several layers. This page names the steps and the vocabulary that keeps them apart (issue #1724), and links to where each step is implemented.

physical plasma state                    n, T, B, q, a, R, I_p(t), ...
        ↓
dimensional characteristic scales        Ω_ci, ρ_i, d_i, λ_mfp, L_T, τ_A, τ_R, ...
        ↓
ratios of scales                         ρ_i/L_T, d_i/a, λ_e/qR, τ_evol/τ_A, ...
        ↓
asymptotic orderings                     ρ_i/L_T ≪ 1, S ≫ 1, ...
        ↓
reduced physical models                  ideal MHD, drift kinetics, gyrokinetics, ...
        ↓
state-specific applicability             SUPPORTED / OUTSIDE / UNASSESSED / NOT_APPLICABLE
        ↓
diagnostic / numerical resolvability     is the structure resolved at all?

Each arrow is a separate step, and none of them is implied by the one before.

Four different kinds of statement

A characteristic scale on its own establishes nothing. $\rho_i = 3$ mm is neither small nor large until it is divided by something, and the answer changes with the divisor (§ Name the comparison scale). A regime name is a reading of orderings. The banana regime is $\nu_* \ll 1$, and the plateau regime has no ordering of its own (Asymptotic orderings, “Ordering contracts”). The neoclassical and NTV regime diagrams of #1111 (ntv_precession_regimes and the collisionality-regime figures in Scientific diagrams) draw regimes the same way: as readings of ratios, with boundaries only where the theory gives one [7]. Applicability is a verdict on a state and belongs to the validation layer. It is never a property of the ordering’s definition.

What exists today

The page distinguishes three states, and says which applies everywhere it names an interface:

• implemented on develop. A function name in code is importable from develop.
• planned under an open issue. Named with its issue; no interface is promised.
• open PR only. None at the time of writing. Interfaces in open PRs are not documented here as stable.

The main gap is #1064 (open): a canonical scale layer (vaft.formula.scales) does not exist yet. Today each scale lives in the domain module that first needed it, listed below. When #1064 lands, the tables below should point at it and the domain functions stay as convention-bearing variants.

Scale taxonomy, by physical role

Scales are grouped by what they mean physically, not by the module that computes them. All functions are in vaft.formula unless the path says otherwise. Units are SI, temperatures in eV.

Frequencies and rates

Not implemented:

• a direct NRL $\nu_{ei}$ or $\nu_{ee}$ frequency function. Use the inverse of a collision time, and say so.
• magnetic-drift precession $\omega_D$ and $E\times B$ precession $\omega_E$ as computed quantities. They are inputs [rad/s] to ntv.ntv_precession_frequency and particle.bounce_harmonic_detuning. The drift velocities (particle.exb_drift_velocity, grad_b_drift_velocity, curvature_drift_velocity) are velocities, not frequencies. vaft.code.transp.profiles.exb_frequency reads $\omega_E$ from TRANSP output.

Angular frequency, ordinary frequency and inverse time are different quantities.

Every angular-frequency function above returns rad/s; the collision times are in s and $\nu_{\rm eff}$ in 1/s. At $B = 0.1$ T, $\Omega_{ce} \approx 1.76\times10^{10}$ rad/s is $f_{ce} \approx 2.8$ GHz. Comparing $\Omega_{ce}$ with a 2.8 GHz bandwidth without the $2\pi$ is an error of 6.3. A period is $2\pi/\omega$, not $1/\omega$: $1/\Omega$ is the time to turn one radian, and that is what timescale_hierarchy plots as the gyration time.

Speeds

Microscopic lengths

These are not universally ordered by magnitude. With $v_t=\sqrt{T/m}$, $d_i/\rho_i = \sqrt{2/\beta_i}$: in a low-$\beta$ tokamak $d_i$ exceeds $\rho_i$, by a factor that depends on the state, and the two meet only at $\beta_i\sim1$. $d_e/\rho_e$ is $\sqrt{2/\beta_e}$ in the same way. $\lambda_D$ is the smallest in a fusion plasma, but not by a fixed factor.

Orbit and parallel-motion scales

The analytic orbit scales carry the large-aspect-ratio ordering $\epsilon \ll 1$. In VEST $\epsilon \approx 0.7$ (the #1629 atlas median), so they are order-of-magnitude estimates there, not results.

Fluctuation and layer lengths

Neither is a property of the equilibrium state alone, so VAFT has no function that returns them. The mode-scope and layer-scope ordering quantities take them as inputs and are UNASSESSED without them.

Profile and equilibrium scales

These need three choices before they are numbers, and VAFT keeps them apart (#551 and #563, closed):

vaft.process.profile_gradients.profile_gradient takes all three explicitly, and its convention presets (tglf, cgyro, gs2, gkw, gene) fix them the way each code does. The table of combinations is in Equilibrium representations, “Profile gradients”. The pure kernel is utils.normalized_gradient_scale_length(x, y, a), which returns $a/L_y$. There is no bare L_n/L_T function on purpose: a gradient length without its coordinate is ambiguous.

Other equilibrium scales: minor radius $a$ and major radius $R_0$ with equilibrium.inverse_aspect_ratio_from_a_R; the magnetic shear length geometry.shear_length_from_q_R0_s (shear itself from equilibrium.shear_from_r_q).

Macroscopic times

These are different physical constructions. One state can have several “times” with the same name, and they must not be swapped:

Scale classes are not a magnitude ordering

A schematic ladder is useful for orientation:

lengths:  Debye → gyro / inertial → orbit / turbulence → profile gradient → minor radius → major radius
times:    gyro motion → orbit / collisions → Alfvénic → mode / wall response → transport → current diffusion → discharge evolution

These are scale classes, not a universal magnitude ordering. The actual order is a property of one state and must be computed from it. vaft.diagram.timescale_hierarchy does exactly that: it evaluates each time from a formula kernel for one illustrative low-field state (only the pulse length is an input) and places it by value, so the hierarchy is a result, not an assumption (Asymptotic orderings, “A worked hierarchy”).

The ladder fails in practice:

• In a low-field device the electron transit frequency can reach the ion gyrofrequency. In VEST, $\omega_{t,e}\approx5\times10^6$ against $\Omega_{ci}\approx10^7$ rad/s (Plasma models §7).
• A short pulse compresses the slow end. In the #1629 atlas the median $\tau_{\rm age}/\tau_R$ is 0.57: current diffusion is not slower than the discharge, so “discharge evolution” is not the last rung.

Name the comparison scale

L is part of an ordering’s definition. It is never universal. The same numerator over different denominators answers different questions:

Every ordering quantity in vaft.validation.orderings.ORDERING_QUANTITIES records its comparison scale explicitly. The issue’s proposed vocabulary maps onto those fields:

The kernels in vaft.formula.ordering take the length as an explicit argument for the same reason. A global length cannot be silently swapped for a layer length.

Similarity coordinates are not orderings

Some dimensionless numbers place a state in similarity space. Others are the small or large parameters a derivation expands in. Some symbols appear in both lists, but the role decides what a value means.

Concretely:

rho_i / a      → machine similarity / finite-size coordinate     (ρ*: equilibrium.rho_star_from_M_T_B_R_epsilon, on a = R_geo ε)
rho_i / L_T    → local FLR / background-scale ordering            (rho_i_over_LTi)
k_perp rho_i   → fluctuation-scale ordering                       (k_perp_rho_i)

The same function can serve both roles. $\epsilon = a/R_0$ is a shape coordinate in a confinement fit and the expansion parameter of the large-aspect-ratio ordering. Sauter’s $\nu_{*e}$ is a regime coordinate in a neoclassical fit and the ordering of the banana-regime contract. The number is identical; the claim is not.

Similarity coordinates in VAFT:

• $\rho_*$: equilibrium.rho_star_from_M_T_B_R_epsilon (Verdoolaege form, on $a$), equilibrium.normalized_larmor_radius_from_M_T_a_Bt ($\rho_i/a$);
• $\nu_*$: neoclassical.electron_collisionality_sauter, neoclassical.ion_collisionality_sauter, equilibrium.nu_star_from_n_T_B_R_epsilon_kappa_I (Verdoolaege form);
• $\beta$, $\beta_N$, $q$: equilibrium.beta_t_from_n_T_B, stability.beta_N_from_beta_a_B0_Ip, equilibrium.q_cyl_from_B_R_epsilon_kappa_I.

Different $\nu_$ definitions [8, 12] differ by order-unity factors. Never compare two without naming both. vaft.process.confinement.dimensionless_confinement_indices turns fitted engineering exponents into $\rho_$, $\beta$, $\nu_*$, $q$ exponents; its docstring says they are not measured dimensionless dependences.

Geometry is not an ordering

Slab, cylindrical and toroidal describe geometry [3]. Local or global, $a/R\ll1$, $\rho/L\ll1$, $k_\parallel/k_\perp\ll1$ and $\omega/\Omega\ll1$ describe scale separation. They are two axes, not one accuracy ladder. “Large aspect ratio” is an ordering, not a geometry, and a local slab may keep coefficients inherited from a realistic toroidal equilibrium (#1062). The full treatment, with geometry_ordering_map, is Geometric approximations, “Two axes, not one hierarchy”.

From orderings to models

Each plasma description is defined by the orderings it adopts and the ones it relaxes. It is not a rung on one fidelity ladder. Plasma models is the reference:

• §1 explains why a single “fidelity” axis fails;
• §2 derives the fluid hierarchy (ideal, resistive, Hall/two-fluid, reduced MHD [5]), drift kinetics, gyrokinetics, particle orbits and kinetic-MHD hybrids;
• §6 says what “kinetic” means in each VAFT name;
• §7 is the compact table of what each family removes and which ordering justifies it.

The comparison axes are:

ρ/L    d_i/L    ω/Ω_i    k⊥ρ_i    k∥/k⊥    collisionality    resistivity    locality    distribution-function dynamics

The ordering contracts of #1627 encode them per model (vaft.validation.orderings.CONTRACTS, drawn as ordering_contract_map). The adapter modes of #1823 say which contracts each VAFT code mode consumes, and in which role (governing, outer_solution, layer_solution, fluid_part, kinetic_response, closure, inherited, transport_coupling). That is vaft.validation.model_contracts.ADAPTER_MODES, drawn as adapter_contract_map (Asymptotic orderings, “Which codes consume which contracts”).

Drift kinetics and gyrokinetics are not one ladder

A guiding-centre or drift-kinetic description assumes fields that vary slowly compared with gyromotion, $\rho/L\ll1$ and $\omega/\Omega\ll1$, on every scale it keeps [6]. A gyrokinetic description keeps the low frequency ordering but allows perpendicular structure at the gyroradius, $k_\perp\rho_i\sim O(1)$, at the cost of ordering the fluctuation amplitude and $k_\parallel/k_\perp$ small [9, 14]. So:

drift kinetic   ≠  a lower-fidelity gyrokinetic model
gyrokinetic     ≠  "more kinetic" MHD

They keep different scale information for different problems. Drift kinetics is the natural description of equilibrium-scale kinetic response (neoclassical transport, NTV, kinetic stabilization). Gyrokinetics is the description of $\rho$-scale turbulence, with transport on a separate, slower scale [10]. Plasma models §2.2–§2.3; the gyrokinetic framework is #1353 (open).

Kinetic GPEC is not on the gyrokinetic branch

Kinetic DCON/GPEC and PENTRC are not on the same branch as CGYRO. Both evolve a gyro-averaged kinetic equation under $\omega/\Omega\ll1$, and gyrokinetics reduces to drift kinetics at long wavelength, but they answer different questions:

MHD perturbed equilibrium
  + kinetic orbit / distribution response
  + bounce / transit / drift resonances
        ↓
kinetic correction to the global plasma response            (DCON kinetic, GPEC kinetic, PENTRC) [15-17]

gyrokinetic microturbulence
        ↓
turbulent transport                                          (CGYRO, TGLF)

They differ on every axis that matters: physical formalism (bounce-harmonic drift kinetics against $\delta f$ gyrokinetics), spatial domain (global against flux-tube local), frequency ordering (precession and bounce resonances against drift-wave frequencies) and the problem solved (response or stability against turbulent flux). vaft.code.formalism.PlasmaFormalism records these as separate fields for exactly this reason (Plasma models §3, §4, §11).

From orderings to applicability

An applicability verdict is assembled from separate assumptions, each with its own evidence. The ideal_single_fluid_mhd contract needs, among others,

and these fail independently. Slow evolution, $\tau_{\rm evol}/\tau_A\gg1$, is not one of them: ideal MHD describes Alfvénic dynamics, $\omega\tau_A\sim1$. Slow evolution is the separate quasi_static_equilibrium contract, which an equilibrium sequence consumes. VAFT never reduces them to ideal_mhd_valid = True. vaft.validation.applicability.evaluate_contract returns a margin and a status for every assumption, names the limiting one, and reports a missing quantity as UNASSESSED. Local gyrokinetic applicability likewise involves $\rho_s/L_T$, $\rho_s/L_n$ and $\rho_s$ over the geometric scale, plus fluctuation orderings where a spectrum is available.

This page defines no verdicts. The evaluation, its statuses and its margins are Credibility and applicability. The state tables are built by vaft.process.ordering_state. The VEST population study is the notebook vest_ordering_and_model_applicability.ipynb (#1629).

One model, two scales

A model can be asymptotically appropriate at one scale and inadequate at another. Tearing is the standard example. Take the #1629 atlas medians for VEST: $d_i/a\approx0.32$ and $S\approx1\times10^5$.

• Globally, $d_i/a\approx0.3$ is below one. The Hall term is not negligible, but single-fluid MHD is not obviously wrong on the device scale.
• The resistive layer is much thinner than $a$. Its width scales as $\delta/a\propto S^{-2/5}$ for the Furth–Killeen–Rosenbluth tearing layer [4] and $S^{-1/3}$ for the resistive internal kink [13], so $\delta/a\sim0.01$–$0.02$ here, up to factors of mode number and shear.
• Inside the layer, $d_i/\delta\sim15$–$30$. The two-fluid ordering fails by more than a decade. A single-fluid resistive layer model is outside its contract, even though the device-scale ordering looked acceptable.

The same holds for resistivity: $S_{\rm global}\gg1$ does not make resistivity negligible in the layer, because the layer is where it is not. This is why VAFT registers layer quantities (ion_skin_depth_over_layer, electron_skin_depth_over_layer, rho_s_over_layer) separately from the global ones, and why the RDCON/RMATCH adapter mode consumes resistive_mhd, which orders $d_i/\delta$ and $\rho_s/\delta$, in the layer_solution role, next to ideal_single_fluid_mhd in the outer_solution role.

Physical scales are not resolution

Whether a structure is resolved is a different question from how big it is. A diagnostic cadence or a grid spacing is never a characteristic scale of the plasma, and this page does not define one.

Comparing the two belongs to an analysis layer above both, for example “is the EFIT cadence shorter than $\tau_{\rm evol}$?” or “does the Thomson chord spacing resolve $L_{T_e}$?”. Not implemented. Only the time-grid description exists today (vaft.process.signal_processing.describe_time_grid). Nothing compares a cadence or a grid with a physical scale yet. When it is built, it is evidence on the numerical or transformation axis of the credibility vocabulary, not a new kind of scale.

Diagram contracts

Four diagrams make the chain visible. Two exist. All of them follow the rule of this page: a magnitude order is drawn only when it comes from values.

Diagrams live in vaft.diagram and are conceptual. Ladders of measured states belong to vaft.plot, with the values from vaft.process. The two planned diagrams wait for #1064, so that they draw canonical scale functions rather than define their own.

Who owns what

vaft.formula.scales           generic dimensional scale primitives                          planned, #1064
vaft.formula.<domain>         convention-bearing scales (neoclassical, wall, equilibrium, ...)   implemented
vaft.formula.ordering         ordering kernels with explicit lengths                          implemented, #1627
vaft.process                  state- and profile-aware construction, coordinate and time alignment
                              (profile_gradients, ordering_state)                            implemented
vaft.validation               ordering registry, contracts, applicability, credibility       implemented, #1627 / #1639 / #1628
vaft.diagram                  conceptual scale and ordering diagrams                          partly implemented
vaft.plot                     scale ladders and ordering spaces of real states                ordering spaces implemented (#1627 §3)

This page consumes those layers. It defines no formula: every expression above is either the definition a linked kernel implements, or a scaling quoted with its source.

References

1. J. D. Huba, NRL Plasma Formulary, Naval Research Laboratory (2019). Plasma and gyrofrequencies, Debye length, gyroradii, inertial lengths, collision rates.
2. S. I. Braginskii, in Reviews of Plasma Physics, Vol. 1, Consultants Bureau (1965), p. 205. Collision times and the collisional fluid ordering.
3. J. P. Freidberg, Ideal MHD, Cambridge University Press (2014). The ideal-MHD ordering and the large-aspect-ratio expansion.
4. H. P. Furth, J. Killeen and M. N. Rosenbluth, Phys. Fluids 6 (1963) 459. Resistive layer width and its Lundquist-number scaling.
5. H. R. Strauss, Phys. Fluids 19 (1976) 134. Reduced MHD.
6. R. D. Hazeltine and J. D. Meiss, Plasma Confinement, Dover (2003). Drift-kinetic ordering and orbit scales.
7. F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48 (1976) 239. Neoclassical regimes and collisionality.
8. O. Sauter, C. Angioni and Y. R. Lin-Liu, Phys. Plasmas 6 (1999) 2834. The $\nu_*$ definitions VAFT uses.
9. E. A. Frieman and L. Chen, Phys. Fluids 25 (1982) 502. The nonlinear gyrokinetic ordering.
10. I. G. Abel et al., Rep. Prog. Phys. 76 (2013) 116201. Multiscale gyrokinetics and the separation of turbulence and transport scales.
11. B. B. Kadomtsev, Sov. J. Plasma Phys. 1 (1975) 295. Dimensionless similarity.
12. G. Verdoolaege et al., Nucl. Fusion 61 (2021) 076006. The $\rho_$ and $\nu_$ forms of the ITPA20 scaling.
13. B. Coppi, R. Galvão, R. Pellat, M. N. Rosenbluth and P. H. Rutherford, Sov. J. Plasma Phys. 2 (1976) 533. The resistive internal kink and its $S^{-1/3}$ layer.
14. J. A. Krommes, Annu. Rev. Fluid Mech. 44 (2012) 175. The gyrokinetic description of microturbulence.
15. M. D. Kruskal and C. R. Oberman, Phys. Fluids 1 (1958) 275. The kinetic energy principle.
16. J.-K. Park, A. H. Boozer and A. H. Glasser, Phys. Plasmas 14 (2007) 052110. Ideal perturbed equilibrium.
17. N. C. Logan, J.-K. Park, K. Kim, Z. Wang and J. W. Berkery, Phys. Plasmas 20 (2013) 122507. NTV from the perturbed equilibrium (PENT).