Normalization

This section documents the normalization conventions used in SPECTRAX-GK and the calibration parameters that scale drift/drive terms for benchmark comparisons.

Canonical normalization contract

SPECTRAX-GK now centralizes benchmark-family normalization values in spectraxgk.normalization via spectraxgk.normalization.NormalizationContract. This is the single source of truth for case defaults:

Canonical per-case normalization contracts
Case key	`rho_star`	`omega_d_scale`	`omega_star_scale`	`diagnostic_norm_default`
`cyclone`	`1.0`	`1.0`	`1.0`	`none`
`etg`	`1.0`	`0.4`	`0.8`	`none`
`kinetic` (`kinetic_itg` alias)	`1.0`	`1.0`	`1.0`	`none`
`tem`	`1.0`	`1.0`	`1.0`	`none`
`kbm`	`1.0`	`1.0`	`0.8`	`none`

These contracts are consumed by benchmark constants for backward compatibility (CYCLONE_OMEGA_D_SCALE, etc.), so existing scripts keep working while all new calibration updates flow through one module.

Dimensionless units

We evolve a dimensionless gyrokinetic system normalized to ion thermal quantities. The parallel streaming term uses the normalized thermal velocity \(v_{th}\) and the flux-tube coordinate \(z\) (theta-like). The perpendicular wave numbers are normalized by the ion gyro-radius \(\rho_i\), while the distribution function and potential use the standard gyrokinetic scaling:

\[\tilde{\phi} = \frac{e \phi}{T_i}, \qquad \tilde{\omega} = \frac{\omega}{v_{th}/R_0}.\]

For the Cyclone base case we take the reference length \(L_{ref}=a\) so that the input gradients are expressed as \(a/L_T\) and \(a/L_n\). With \(R_0 = R/a\), this means \(R/L_T = (R_0)\,(a/L_T)\) and similarly for \(R/L_n\).

Kinetic species conventions

SPECTRAX-GK supports multi-species kinetic systems. Species-dependent arrays carry the charge, mass, density, temperature, and gradient inputs:

charge_sign: \(Z_s\) (e.g., \(+1\) for ions, \(-1\) for electrons).
temp / mass / density: normalized to the reference species.
tz: \(Z_s / T_s\) coupling used in the field terms.
R_over_LTi / R_over_Ln: normalized gradients for each species.

For adiabatic closures, tau_e provides the ratio between the kinetic species temperature and the Boltzmann species temperature.

Field-aligned grid parameters

For the Cyclone base case we use a field-aligned grid with:

\[y_0 = 20,\qquad n_\theta = 32,\qquad n_{period} = 2.\]

In SPECTRAX-GK these map to GridConfig(y0=20, ntheta=32, nperiod=2), which sets:

\[L_y = 2 \pi y_0,\qquad z \in [-\pi Z_p, \pi Z_p),\qquad Z_p = 2 n_{period} - 1.\]

The reduced scan tables and regression tests use Nx=1, Ny=24, Nz=96 on this grid to match the discrete ky set used in the reference CSV.

Spectral grids

SPECTRAX-GK’s explicit benchmark integrator uses the compressed Fourier conventions required by the tracked reference data. The perpendicular wave numbers are defined as

\[k_x = \frac{n_x}{x_0}, \qquad k_y = \frac{n_y}{y_0},\]

with x0 = Lx / (2π) and y0 = Ly / (2π). The parallel wave number is

\[k_z = \frac{n_z}{Z_p},\]

where \(Z_p\) sets the field-line length (\(z \in [-\pi Z_p, \pi Z_p)\)), and \(k_z\) is defined without the gradpar factor. These definitions are implemented by spectraxgk.grids.build_spectral_grid and are consistent with GX’s kInit kernel.

The midplane index used by the reference growth-rate diagnostic corresponds to z_index = Nz//2 + 1, matching the audited benchmark kernel logic when Nz > 1.

Perpendicular normalization

The reference diagnostic contract defines the perpendicular metric as \(k_\perp^2/B^2\) before applying the Laguerre gyroaverage. To match that convention in SPECTRAX-GK:

kperp2_bmag = True (include the \(B^{-2}\) factor in \(k_\perp^2\))
bessel_bmag_power = 0 (no extra \(B\) scaling inside the Bessel argument)

The Cyclone base case defaults follow this benchmark setting, and the compare_gx_rhs_terms.py comparison tool assumes the same normalization.

Sign conventions

The growth-rate fitting in spectraxgk.analysis.fit_growth_rate() assumes

\[s(t) \sim \exp((\gamma - i \omega)\, t),\]

so:

gamma > 0 indicates instability.
omega is obtained from the negative phase slope.

This is consistent across time-integration and Krylov post-processing paths. For scan tables and figures, values are reported in the same sign convention as the solver output unless an explicit diagnostic normalization is requested.

Normalization parameters

The linear operator exposes three normalization parameters that influence the drift/drive terms:

rho_star: scales \(k_x\) and \(k_y\) in the drift and drive terms.
omega_d_scale: scales curvature/grad-\(B\)/mirror couplings.
omega_star_scale: scales the diamagnetic drive.

In code, rho_star multiplies the Fourier grids inside spectraxgk.linear.build_linear_cache(), while omega_d_scale and omega_star_scale enter directly in spectraxgk.linear.linear_rhs_cached().

Diagnostic normalization mode

Benchmark runners expose diagnostic_norm and route it through spectraxgk.normalization.apply_diagnostic_normalization:

none: return raw solver (gamma, omega).
gx / rho_star: multiply reported (gamma, omega) by rho_star.

This affects reporting only; it does not alter the RHS/operator.

The unified runtime schema defaults to diagnostic_norm = "gx" so that out-of-the-box reports match the tracked benchmark normalization. Set diagnostic_norm = "none" in the TOML or runtime config to recover raw solver outputs.

Diagnostic scaling

The reference diagnostics apply fixed factors in a few places that depend on the storage convention (e.g. real-FFT nyquist handling or per-unit-time damping). The runtime schema therefore exposes light-weight diagnostic scale factors:

flux_scale: multiplicative factor applied to the reported heat/particle fluxes (default 1.0 for the tracked comparison convention).
wphi_scale: multiplicative factor applied to Wphi (default 1.0; the Cyclone nonlinear comparison config uses 1.155).

These are reporting-only knobs; they do not alter the RHS/operator. They are intended to document exact comparison settings used for benchmark plots.

The reference end-damping defaults are damp_ends_amp = 0.1 and damp_ends_widthfrac = 0.125. The damping kernel interprets damp_ends_amp directly as a rate; the runtime no longer rescales it by the timestep.

Defaults (model parameters):

rho_star = 1.0 (model default)
omega_d_scale = 1.0 (model default)
omega_star_scale = 1.0 (model default)

These parameters are surfaced in the regression tables so that future normalization refinements can be tracked in a reproducible way.

Programmatic usage

from spectraxgk.normalization import get_normalization_contract

contract = get_normalization_contract("etg")
# contract.omega_d_scale == 0.4
# contract.omega_star_scale == 0.8