Asymptotic orderings

Some dimensionless numbers say how well a plasma performs, or where it sits in similarity space ($\beta_N$, $\rho_$, $\nu_$). Others are the small or large parameters a model’s derivation expands in. A Lundquist number tests whether resistive diffusion is slow compared with Alfvénic dynamics. $d_i/L$, $\rho_i/L$, a Knudsen number or $\Omega\tau$ each test a different reduction. This page covers the second kind: asymptotic ordering parameters (#1627). The question they answer is which reduced models the scale separation of a given state justifies.

The kernels are in vaft.formula.ordering. Their inputs come from VAFT’s Alfvén speed, Spitzer resistivity, gyrofrequency and Larmor radius formulas, and the inertial length calls the plasma frequency. The characteristic length, the resistivity and the collision time are always explicit arguments, so a global scale cannot be silently swapped for a layer scale.

How a characteristic scale becomes an ordering, why the comparison length must be named, and how orderings differ from similarity coordinates and from resolution is Characteristic scales and orderings.

The parameters and what they test

Some distinctions must not be blurred:

• A large global $S$ does not make resistivity irrelevant in a tearing layer. The layer’s own $S$, and $d_i/\delta$, $\rho_s/\delta$, decide that.
• Global $\rho_*$ is a similarity coordinate. Local $\rho_i/L_T$ is an FLR ordering. They are not interchangeable.
• $\tau_{evol}/\tau_A \gg 1$ is physics. It says nothing about whether an EFIT run converged, and convergence says nothing about it.
• $Kn$ and $\Omega\tau$ are independent. A plasma can be strongly magnetized and collisionless along the field at the same time.

A worked hierarchy

timescale_hierarchy evaluates the hierarchy of one illustrative low-field spherical-tokamak state rather than assuming it. Every time comes from a formula kernel; only the pulse length is an input.

• Clearly ordered: $S$ and $\tau_{evol}/\tau_A$, both about $10^4$.
• Of order one, so not ordered:
  • $\tau_{pulse}/\tau_R \approx 0.7$. Resistive relaxation has to be checked, not assumed either way. $\tau_R$ carries no geometric factor, and the slowest cylindrical mode decays $j_{01}^2 \approx 5.8$ times faster.
  • $d_i/a \approx 0.3$, so Hall corrections are not globally negligible.

The axis also places the sound crossing time $a/c_s$ and the inverse thermal-ion transit and bounce frequencies (#1739). They show where neoclassical and drift-kinetic dynamics sit between $\tau_A$ and the evolution time. Drift precession has no formula kernel, and $\tau_E$ needs an energy inventory the state does not carry, so neither is drawn. spatial_scale_orderings places the lengths of the same state on one axis: the Larmor radii, $\rho_s$, $d_i$, the banana width, an illustrative gradient length, $a$, $R_0$ and $qR_0$.

These numbers come from an illustrative state, not a measured discharge. Answering the same questions across the VEST database is the ordering atlas of #1629.

Ordering contracts

Whether an ordering holds depends on the model that assumes it. The contracts are registered in vaft.validation.orderings.CONTRACTS as ApproximationContract objects of vaft.validation.applicability (#1639). They are evaluated with evaluate_contract for one state and evaluate_population for a table, whose columns are the names in ORDERING_QUANTITIES.

The quantities are grouped by the scale they are taken on, because the same ratio means different things on different scales. A global $d_i/a$, a mode’s $k\,d_i$ and a tearing layer’s $d_i/\delta$ are three rows: $d_i/a$ can be small while $d_i/\delta \sim 1$, and single-fluid MHD then fails in the layer.

A contract lists only what its model requires. An empty cell means the model does not order that quantity, which is different from requiring it to be large. The parallel Knudsen number $\lambda/qR$, the inverse of $\hat\nu$ up to an order-one convention factor, is shared by the Braginskii closure and the Pfirsch–Schlüter regime: the collisional closure is the Pfirsch–Schlüter ordering, and it fails in the banana regime of a hot core. The reduced-MHD contracts list only the reduction’s own orderings, so evaluate them together with the ideal or resistive MHD contract.

Every threshold is order unity. The result is a continuous margin per ordering, $m = \mp\log_{10}x$, and a status per contract (SUPPORTED, OUTSIDE, UNASSESSED, NOT_APPLICABLE), never a single valid/invalid flag. A quantity a state does not carry is UNASSESSED, never a violation. Signed ratios (Mach numbers, the pressure anisotropy) enter as magnitudes. An exactly zero value, as in a static or isotropic state, has no logarithmic margin and is also UNASSESSED.

This covers the fluid-to-kinetic core of the hierarchy, not every subsidiary ordering of every derivation. Full Vlasov–Maxwell kinetics assumes none of these orderings and has no contract. A gyrofluid shares the gyrokinetic orderings and differs only in its closure. The plateau regime between banana and Pfirsch–Schlüter has no ordering of its own.

Ordering spaces and margins

Two orderings read together form an ordering space. Its lines are where an expansion parameter reaches one. They are not empirical stability or operating boundaries. The spaces are registered in vaft.validation.orderings.ORDERING_SPACES:

import pandas as pd
from vaft.process.ordering_state import ordering_table
from vaft.plot.ordering_space import ordering_space, ordering_margins_plot

# One row per state (illustrative VEST-like inputs); ordering_table adds the ordering columns.
inputs = pd.DataFrame({
    "a_m": [0.25, 0.25, 0.25], "r_geo_m": [0.45, 0.45, 0.45], "b_t_T": [0.15, 0.2, 0.25],
    "n_e_m3": [5e18, 1e19, 2e19], "t_e_eV": [20.0, 50.0, 100.0], "z_eff": [2.0, 2.0, 2.0],
    "ln_lambda": [12.0, 12.0, 12.0], "beta_t": [0.01, 0.02, 0.03],
    "efit_quality": ["good", "good", "poor"],
})
states = ordering_table(inputs).join(inputs[["efit_quality"]])
# One row per flux-surface point, with the profile orderings.
points = pd.DataFrame({"rho_i_over_LTi": [0.02, 0.05, 0.2, 0.8]})

fig, ax = ordering_space(states, "resistive_two_fluid", group="efit_quality")
fig, ax = ordering_margins_plot(states, points)   # global orderings per state, profile ones per point

ordering_space draws a population on log axes. It shows:

• the order-unity line of each axis;
• the region where both orderings hold, shaded;
• dashed lines one and two decades into the permitted side;
• the four quadrants, named by the regime each side means.

A quadrant name describes only these two orderings. “Ideal, single-fluid” says nothing about the other orderings of the ideal-MHD contract. A row whose value is missing or not positive is counted as unassessed and left off, never placed at a guessed position.

ordering_margins_plot(states, points) shows, for each ordering, the margin $m = \mp\log_{10}(x/x_0)$ of every state or point, its median, the fraction with $m > 0$ and the number assessed. A global or time-history ordering is read from states, once per state. Any other ordering is read from points, once per flux-surface point. So $n$ and the fraction match vaft.process.ordering_state.ordering_margins on the same data. A row without one quantity is left out of that ordering only. Each ordering is drawn against its contract threshold $x_0$ (ordering_threshold), which is order unity today. The background bands are MARGIN_LABELS:

The bands name the scale separation. They are not an applicability status: that stays the two-sided SUPPORTED / OUTSIDE of evaluate_contract, which has no “marginal” on purpose. A margin of exactly zero, a quantity at its threshold, falls in “outside”, as it is OUTSIDE there. The continuous margin is the result, and margin_label only names its band.

Which codes consume which contracts

A contract says what a model assumes. A run’s vaft.code.formalism.PlasmaFormalism record (#1727) says which model the run used. vaft.validation.model_contracts joins the two (#1823). Every mode with its record and its contracts is tabulated, generated from the registry, in Plasma models: a guide, §9.

contracts_for(formalism) maps a record to the contracts it consumes, as (contract, role) pairs. It reads the record’s fields as they are. The role says how the assumption enters the calculation. It is not a statement about how much of the plasma the assumption covers:

How to read the figure:

• The figure is deliberately simple. The validity conditions behind each cell are listed below, not drawn.
• An empty cell means no registered contract. It does not mean the calculation is independent of that physics. A part of a formalism that no contract covers is returned as a reason, never as a guessed contract:
  • test-particle orbit orderings (NUBEAM);
  • full-$f$ gyrokinetics;
  • a reduced model’s own closure or fit.
• Governing vs inherited. CGYRO solves gyrokinetics, so the gyrokinetic contract governs it. TGLF and TGLF-NN only inherit it. Their gyro-Landau-fluid closure, saturation rule and, for TGLF-NN, the training domain add assumptions that no ordering contract states. A gyrokinetic ordering that holds is therefore necessary for TGLF, not sufficient.

Validity conditions by mode. All modes are checked against the adapter code.

Gyrokinetic–transport separation is opt-in. A standalone flux-tube run, linear or nonlinear, imposes a fixed background. Comparing its fluxes with experiment still assumes $\tau_{transp}/\tau_{turb} \gg 1$. The δf ordering makes turbulence-driven profile evolution $\rho_*^{-2}$ slower than the turbulence. It says nothing about evolution driven by the current ramp or actuators, which dominates VEST’s ~10 ms discharges. The registry books the separate contract only when the run’s fluxes drive a transport solver, as in a PORTALS-type flux-matching loop (#1588, not yet in VAFT). This is a bookkeeping choice. The record has no field that says so. A workflow therefore passes contracts_for(..., transport_coupled=True), or sets AdapterMode.transport_coupled. No VAFT mode sets it today, so the figure has no coupling column.

Sources and checks:

• ADAPTER_MODES lists the modes of VAFT’s adapters with their formalisms, from the audit in Plasma models.
• CGYRO’s two modes are the records its own plasma_formalism builds.
• A test checks the GPEC-suite modes field by field against the records vaft.code.gpec resolves from the packaged namelists.
• Modes VAFT does not run are not listed. RDCON’s RPEC path is one: rdcon.in sets coil%rpec_flag=f, even though the resolver can classify it.
• The modes the audit leaves unresolved, today the GPEC threshold layer models, are listed with no record and a reason.
• Every physics adapter in vaft/code has a mode. A test enforces this. FLARE traces field lines through a given equilibrium and perturbation, and solves no plasma model. MITIM is a driver: the NEO and TGLF runs it launches are those modes, and its PORTALS loop (#1588) will be the first transport_coupled workflow. Both are listed with a reason and no record.

advisory_ordering_check(formalism, state, points) evaluates those contracts on a run’s input:

• state is the global and time-history quantities of vaft.process.ordering_state;
• points is its profile quantities.

A contract with a local ordering is evaluated at every point, and the most constraining point is reported. Each result goes through evaluate_contract and is placed on the credibility applicability axis with as_evidence. The statuses are the four of applicability, with no “marginal”. A quantity the input does not carry is UNASSESSED, never a pass. This covers a mode’s $k_\perp\rho_i$ without mode numbers, and a layer width. The check is advisory: it reports and never blocks a run.

Recorded in run provenance. ordering_provenance(formalism, state, points) reduces the check to a strict-JSON record. It lists the contracts consumed with their roles, statuses, limiting orderings, margins and point counts. A missing margin is null. Three adapters store it, one per family:

The state is advisory. It never enters EFIT’s scientific identity and never changes a namelist or an input file. A failure inside the check is recorded as an error entry; it never costs a run its result. For CGYRO, a finite caller value takes precedence over a derived one, and a caller’s NaN does not hide it. If extra_parameters override species keys (a gradient scan), nothing is derived from the staged input, and the record says why. State keys that are not ordering quantities are listed under ignored.

Building the state from an ODS. vaft.process.ordering_state.ordering_inputs_from_ods(ods, t) returns state, points and a provenance record for one equilibrium slice. It matches that slice to t, and the core_profiles slice to the equilibrium, by time within 0.5 ms, never by index. $Z_{\rm eff}$ comes from the caller or from the stored profile, never from an assumption; without it the resistive orderings stay UNASSESSED. The plasma age needs the caller’s onset time. The kinetic EFIT chain passes KineticEFITConfig.ordering_state to every PLASMA-scale run’s EFITConfig.

Not yet wired. No production workflow passes ordering_state yet, so a routine EFIT or GPEC record lists the consumed contracts with every ordering UNASSESSED until its stage calls the builder (#1823). CGYRO’s two local quantities are in GACODE’s units: $\rho_s$ is deuterium-normalised in $B_{unit}$. That is an order-one difference from the local field, inside the order-unity thresholds.

Known gap. The drift-kinetic contract orders $\omega/\Omega_{ci}$ and $k_\perp\rho_i$. A steady axisymmetric solve (NEO) or a zero-frequency one (DCON kinetic) has neither as an input, so that contract stays UNASSESSED for those modes. A NaN at a profile point never hides a finite state value. The reported point is the DataFrame’s index label when points is a DataFrame.

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