Normalizations and units

SFINCS v3 (and dkx) mixes physical quantities (\(n_s\), \(T_s\), …) with dimensionless normalized quantities written with hats (BHat \(=\hat B\), …). This page collects the normalization conventions used throughout the code and the sfincsOutput.h5 fields. The drift-kinetic equation that uses them is derived in Physics reference: the radially local drift-kinetic model.

Reference scales and hats

All hatted fields are normalized to reference scales: lengths to \(\bar R\), magnetic field to \(\bar B\), so that

\[\hat B = \frac{B}{\bar B}, \qquad \hat\psi = \frac{\psi}{\bar B\,\bar R^2}, \qquad \hat r = \frac{r}{\bar R},\]

and the Boozer flux functions are \(\hat G = \langle B_\zeta\rangle/\bar B\bar R\), \(\hat I = \langle B_\theta\rangle/\bar B\bar R\) (GHat, IHat).

The drift-kinetic ordering parameters

Three scalars set the physical regime and are supplied through the namelist as Delta, alpha, nu_n:

\[\Delta = \frac{\bar m\,\bar v}{Z\,e\,\bar B\,\bar R} \;\sim\; \frac{v_{\mathrm{th}}}{\Omega\,\bar R}, \qquad \alpha = \frac{e\,\bar\Phi}{\bar T}, \qquad \nu_n = \frac{\bar\nu\,\bar R}{\bar v}.\]

  • \(\Delta\) (Delta) — drift/gyroradius ordering parameter; multiplies every drift term and every radial flux.

  • \(\alpha\) (alpha) — potential-to-energy normalization; appears in the \(E\times B\) drive, the \(\Phi_1\) Boltzmann factor, and quasineutrality.

  • \(\nu_n\) (nu_n) — collision-frequency normalization multiplying the collision operator.

All three are written verbatim into sfincsOutput.h5 (Outputs (HDF5, NetCDF4, and NPZ)).

Velocity-space coordinates

The drift-kinetic equation is solved on the 4D phase space \((\theta,\zeta,x,\xi)\):

  • \(\theta,\zeta\) — Boozer angles (or the appropriate substitutes for the geometry scheme; see Geometry models and loading);

  • \(x_s = v/\sqrt{2T_s/m_s}\) — normalized speed, discretized on the Landreman–Ernst grid over a finite interval \(0\le x\le x_{\max}\);

  • \(\xi = v_\parallel/v\) — pitch-angle cosine, expanded in Legendre modes.

The species Maxwellian is \(f_M = n_s\,(m_s/2\pi T_s)^{3/2}\,e^{-x_s^2}\) and the distribution is split \(f_s=f_{s0}+f_{s1}\) (see Physics reference: the radially local drift-kinetic model).

Flux-surface averages

Constraints and diagnostics use the Jacobian-weighted flux-surface average

\[\langle g\rangle = \frac{\int d\theta\,d\zeta\; g(\theta,\zeta)\,\hat D(\theta,\zeta)^{-1}} {\int d\theta\,d\zeta\; \hat D(\theta,\zeta)^{-1}},\]

evaluated with the quadrature weights thetaWeights / zetaWeights and the geometry factor DHat. Two derived scalars appear throughout: \(\hat V' = \sum_{ij} w^\theta_i w^\zeta_j/\hat D_{ij}\) (VPrimeHat) and \(\langle\hat B^2\rangle\) (FSABHat2).

Radial coordinates

dkx supports four radial labels, selected by inputRadialCoordinate for the flux surface and inputRadialCoordinateForGradients for the profile gradients:

\[\psi \;(\texttt{=0}), \quad \psi_N = \psi/\psi_a \;(\texttt{=1}), \quad \hat r \;(\texttt{=2}), \quad r_N = r/a \;(\texttt{=3}), \quad E_r \;(\texttt{=4, gradients}).\]

Each per-species flux is emitted in all four coordinate variants (*_psiHat, *_psiN, *_rHat, *_rN); the conversion multiplies the native \(\nabla\hat\psi\) flux by \(d(\text{coord})/d\hat\psi\) (Outputs (HDF5, NetCDF4, and NPZ)).

Radial electric field

The equilibrium radial electric field is \(E_r=-d\Phi_0/dr\). The operator coefficients are written in terms of \(d\hat\Phi/d\hat\psi\) (the Er, dPhiHatdpsiHat, dPhiHatdrHat, … inputs are inter-converted through the radial-coordinate factors). The ambipolar solve drives Er directly (inputRadialCoordinate = 4); see Physics reference: the radially local drift-kinetic model and dkx.er.

Monoenergetic collisionality and electric field

For monoenergetic (RHSMode = 3) runs the physical collisionality and electric field are supplied as nuPrime and EStar instead of nu_n and dPhiHatdpsiHat:

\[\nu' = \frac{(\hat G + \iota\,\hat I)\,\hat\nu}{v\,\hat B_0}, \qquad E^\* = \frac{\hat G}{\iota\,v\,\hat B_0}\,\frac{d\hat\Phi}{d\hat\psi},\]

with \(\hat B_0\) the \((0,0)\) Fourier mode of \(\hat B\). In this mode the speed grid collapses to a single node at \(x=1\), and nuPrime / EStar are written to sfincsOutput.h5 alongside the \(2\times2\) transportMatrix. For the full upstream unit derivations see the SFINCS technical notes cited from Upstream SFINCS sources and primary literature.