Reduced representations

VAFT holds many reduced descriptions of one plasma state: $q(\rho) \to q_{95}$, $B_p \to l_i$, $\langle p\rangle \to \beta$, and $T, B, a \to \rho_*$. Each output may be a scalar, but they are different operations. The first extracts a feature, the second is a quadratic integral, and the third and fourth are normalisations. The third also needs a volume average first. This page classifies them on four independent axes:

Dimensionless is not 0-D. A local $\nu^*(\rho)$ is a dimensionless_normalization with a profile_1d output, and a global $\beta$ is one with a scalar_0d output. Likewise, $l_i$ is a quadratic integral of $B_p$, not a moment of $j_\phi$.

reduced_representation_hierarchy_compact() draws the same axes and arrows with only the symbols in each box, for slides and overview pages.

Declaring it

The formula’s docstring is the single source of truth. A formula declares its place in the taxonomy with a Reduction section:

Reduction
---------
input: field_2d
output: scalar_0d
kind: quadratic_integral
locality: global
role: global_descriptor

The vocabulary lives in vaft/formula/_taxonomy.py. The catalog validates every value against it:

• a missing, repeated or unknown key is an error;
• so is a value outside the vocabulary;
• so is a dimensionless_normalization, similarity_transform or similarity_coordinate whose return unit is not -.

vaft.formula.describe(name).reduction exposes it. list_formulas(reduction_kind=..., locality=..., role=..., output_representation=...) filters on it. The generated catalog snapshot carries it as reduction, which is where the tables below come from. Generic numerical helpers carry no Reduction section, because the taxonomy says nothing useful about them.

Families

Each graph is the relation metadata in vaft.formula._taxonomy.REDUCTION_FAMILIES, laid out by vaft.diagram.reduction_graph. When an edge names a formula, its reduction kind is read from that formula’s catalog entry, so the figure cannot disagree with the docstrings. A dashed edge is a step VAFT performs elsewhere, such as a flux-surface average or a profile feature, and states its own kind.

Current and $q$

Both $B_p$ and $j_\phi$ derive from the poloidal flux. $B_p = |\nabla\psi|/R$, and Ampère’s law in Grad–Shafranov form is $\Delta^*\psi = -\mu_0Rj_\phi$. $B_\theta(r)$ is the same poloidal field, flux-surface averaged into the cylindrical profile that $q$ is computed from.

Pressure and energy

Kinetic profiles

Dimensionless similarity

The $\nu_$ and $\rho_$ here are the Verdoolaege engineering forms. The other definitions VAFT carries are tracked in #353 and are not declared equivalent.

Generated tables

By spatial mapping

field 2d + scalar 0d → scalar 0d

field 2d → scalar 0d

profile 1d → profile 1d

profile 1d → scalar 0d

scalar 0d → scalar 0d

By reduction kind

closure

differential

dimensionless normalization

empirical scaling

feature extraction

integral

normalization

quadratic integral

similarity transform

By locality

flux surface local

global

By physical role

closure input

closure output

global descriptor

profile descriptor

regime coordinate

similarity coordinate

stability coordinate

state coordinate