Theory

For a full operator-level derivation tied to the implemented code paths (normalization, Hermite-Laguerre projection, field equations, and growth-rate diagnostics), see Linear Model And Derivations. For the explicit implemented operator set, collisions, hypercollisions, nonlinear brackets, and parameter-to-source mapping, see Operators And Terms.

Gyrokinetic ordering

GKX targets the low-frequency, strongly magnetized regime where the characteristic fluctuation frequency is small compared to the ion cyclotron frequency. In this limit, the phase-space dynamics can be reduced to a five-dimensional gyrokinetic system for the non-adiabatic part of the distribution function. Classic derivations of the gyrokinetic equation can be found in Frieman & Chen (1982) and Antonsen & Lane (1980). [FC82] [AL80]

Flux-tube model

We employ a field-aligned, local flux-tube model in which the perpendicular spatial dependence is represented spectrally and the parallel coordinate is resolved along a field line. This approximation underlies the Cyclone base case benchmark commonly used in gyrokinetic validation studies. [Dimits00]

The default boundary condition is a linked (twist-and-shift) flux tube, so the parallel derivative couples Fourier modes across adjacent \(k_x\) indices. GridConfig(non_twist=True) selects the non-twisting flux tube (NTFT): the linear cache shifts \(k_x\) along the field line by \(\delta k_x = k_y f_{twist}(z) + \rho_* m_0/x_0\), with \(f_{twist} = \hat{s}\,g_{ds21}/g_{ds22}\) and \(m_0\) the nearest-integer link index, in both \(k_\perp\) and the drifts (gkx.operators.linear.cache_builder._build_ntft_kperp_and_drift_arrays).

Hermite-Laguerre velocity space

The perturbed distribution is expanded in a Hermite (parallel velocity) and Laguerre (magnetic moment) basis. For a single species, the expansion is

\[\frac{g(\mathbf{k}, \theta, v_\parallel, \mu)}{F_M} = \sum_{\ell=0}^{N_\ell-1} \sum_{m=0}^{N_m-1} G_{\ell m}(\mathbf{k}, \theta) \mathcal{L}_\ell(\mu B/T)\mathcal{H}_m(v_\parallel/v_{th}),\]

where the normalized velocity basis includes the chosen Laguerre sign and Hermite normalization. Its projected gyroaverage coefficient, not the Laguerre polynomial evaluated at the wavenumber, is

\[J_\ell(b) = e^{-b/2}\frac{(-b/2)^\ell}{\ell!},\]

where \(b = k_\perp^2 \rho^2\). This Laguerre-Hermite formulation is detailed by Mandell, Dorland & Landreman (2017). [MDL17]

Field solve and gyrokinetic variable

GKX supports electrostatic and electromagnetic linear closures. For electrostatic runs, quasineutrality is solved in Fourier space for \(\phi\), with an optional adiabatic response controlled by \(\tau_e = T_i/T_e\):

\[\left(\tau_e + \sum_s \frac{Z_s^2 n_s}{T_s}\left[1-\sum_{\ell} J_{\ell}^2\right]\right) \phi = \sum_s Z_s n_s \sum_{\ell} J_{\ell} G_{\ell, m=0}.\]

For Boltzmann electrons the \(k_y = 0\) response is \(\tau_e(\phi - \langle\phi\rangle)\), with \(\langle\cdot\rangle\) the flux-surface average, so zonal potential carries no electron response. Boltzmann ions (ETG runs) respond to the full \(\phi\). Every field-solve route, serial and sharded, applies the same rule.

Electromagnetic runs solve the coupled quasineutrality/perpendicular-Ampere system for \((\phi, B_\parallel)\) and then obtain \(A_\parallel\) from parallel Ampere’s law. The gyrokinetic variable is

\[H_{\ell m} = G_{\ell m} + \frac{Z_s}{T_s}\,J_\ell \phi \, \delta_{m0} - \frac{Z_s v_{th,s}}{T_s}\,J_\ell A_\parallel \, \delta_{m1} + J_{\ell}^{B}\,B_\parallel \, \delta_{m0},\]

with \(J_{\ell}^{B} = J_{\ell} + J_{\ell-1}\). These relations match the Laguerre-Hermite pseudo-spectral form used in the gyrokinetic literature.

The nonzonal field-system test in tests/unit/operators/test_terms_fields.py assembles a separate dense \(3\times3\) system from GX (arXiv v3), (32)–(34) at \(B=1\). It uses two kinetic species with unequal temperatures/masses, finite FLR, and separate density/current/perpendicular-moment excitations; float32/64 values and normalized residuals must agree. Variable-\(B\) cases test GKX’s additional \(B^{-2}\) perpendicular-Ampere convention only: its physical normalization remains unqualified by those paper equations alone. The independent physical-energy identification below supplies the additional field normalization check. Zonal/gauge tests remain separate.

The companion geometry-to-field test checks the constant-\(B\), nonzonal quadratic implied by GX (arXiv v3), (12), (16), and (32)–(34). For the retained basis it independently forms

\[S_\phi=\sum_{s\ell}n_sZ_sJ_\ell G_{s\ell0},\quad S_A=\sum_{s\ell}n_sZ_sv_{th,s}J_\ell G_{s\ell1},\quad S_B=\sum_{s\ell}n_sT_sJ_\ell^B G_{s\ell0},\]

and

\[F_{\rm src}=\frac12\sum_{s\ell m}n_sT_s|G_{s\ell m}|^2 +\frac12\operatorname{Re}(\phi^*S_\phi-A_\parallel^*S_A+B_\parallel^*S_B).\]

Howes et al. (2006), (B19)–(B20) identify the physical fluctuation energy as particle entropy plus magnetic energy. Applying quasineutrality and the corrected local-\(B\) field equations in published GX, (2.12) and (4.22)–(4.25) gives, mode by mode,

\[F_{\rm phys}=\frac12\sum_{s\ell m}n_sT_s|H_{s\ell m}|^2 -\frac12\sum_s\frac{n_sZ_s^2}{T_s}|\phi|^2 +\frac{k_{\perp,N}^2|A_\parallel|^2+B^2|\mathtt{bpar}|^2} {\beta_{\rm ref}}.\]

The fields here are dimensionless. GX Appendix A, Tables 4–5 normalize \(k_\perp\) by \(\rho_{\rm ref}^{-1}\), \(A_\parallel\) by \(\rho_*\rho_{\rm ref}B_N\), and \(\delta B_\parallel\) by \(\rho_*B_N\); GKX stores \(\mathtt{bpar}=\delta B_\parallel/(\rho_*B(z))\). Hence \(\delta B_\perp/(\rho_*B_N)=k_{\perp,N}A_\parallel\) and \(\delta B_\parallel/(\rho_*B_N)=B\,\mathtt{bpar}\). Together with \(\beta_{\rm ref}=8\pi n_{\rm ref}T_{\rm ref}/B_N^2\) and \(v_{th}=\sqrt{T/m}\), this gives the two \(1/\beta_{\rm ref}\) magnetic terms above (and the \(\beta_{\rm ref}/2\) in Ampere’s law), with no hidden \(\sqrt{2}\) factor.

For the \(B^{-2}\) cache convention tested at varying \(B\), \(k_{\perp,N}^2=\mathtt{kperp2}\,B^2\). The test constructs this expression independently and checks \(F_{\rm src}=F_{\rm phys}\) first at constant \(B=1\), then with the varying-\(B\) volume measure; dropping either magnetic \(B^2\) factor must fail. The constant-\(B\) check exercises both cache conventions but cannot distinguish them. This is a per-mode finite-FLR, nonzonal, periodic identity; zonal/gauge and general multi-mode Hermitian weighting remain open. Within that envelope the componentwise JAX convention is \(\operatorname{conj}(\nabla_G F_{\rm src})=n_sT_sH_s\). The symmetric truncated Hermite ladder and skew-adjoint periodic derivative also give the tested streaming-only identity

\[\operatorname{Re}\sum_{s\ell mz}n_sT_sH_{s\ell m}^* (\partial_tG_{s\ell m})_{\rm stream}=0.\]

This excludes curved-geometry, drive, collision, boundary-damping and nonlinear terms. The runtime Wg + Wphi + Wapar diagnostic is a distinct monitoring quantity and contains no \(B_\parallel\) contribution.

For varying \(B\), the companion term-level gate follows Mandell et al. (2018), (4.4)–(4.6): it contracts streaming plus mirror forcing with \(n_sT_sH_s^*\) and the field-line volume weight \(J(z)\propto 1/(|\mathrm{gradpar}|B)\). The periodic boundary contribution vanishes, and the finite-resolution defect converges once aliased products are resolved. Differentiating the reduced source quadratic with its self-consistent fields gives the same weighted \(n_sT_sH_s^*\) contraction. With equilibrium gradients, collisions, hyper-dissipation, and end damping disabled, the gate also contracts the complete assembled linear right-hand side: streaming, mirror, curvature, and grad-\(B\). Every retained term is nonzero, while wrong volume/sign and missing-imaginary-unit controls produce nonzero exchange. The physical identification and variational identity do not establish a full budget: nonlinear transfer, heat-flux normalization, sources, sinks, and time-discretization remain outside this instantaneous gate.

Linear gyrokinetic operator

In the linear model, the Hermite-Laguerre moments evolve according to a drift/mirror operator,

\[\frac{\partial G_{\ell m}}{\partial t} + v_{\mathrm{th}}\,\mathcal{L}_m[H] + v_{\mathrm{th}}\,b^\prime(\theta)\,\mathcal{M}_{\ell m}[H] = -i Z/T\,\bigl(c_v \mathcal{C}_m[H] + g_b \mathcal{G}_\ell[H]\bigr) + i k_y \phi \,\mathcal{D}_{\ell m},\]

where \(\mathcal{L}_m\) is the Hermite streaming ladder and \(b^\prime(\theta)\) is the parallel magnetic field gradient used in the mirror force. The curvature (cv) and grad-\(B\) (gb) drift couplings are encoded in \(\mathcal{C}_m\) and \(\mathcal{G}_\ell\). Explicitly,

\[\mathcal{C}_m[H] = \sqrt{(m+1)(m+2)} H_{\ell, m+2} + (2m+1) H_{\ell m} + \sqrt{m(m-1)} H_{\ell, m-2},\]

\[\mathcal{G}_\ell[H] = (\ell+1) H_{\ell+1, m} + (2\ell+1) H_{\ell m} + \ell H_{\ell-1, m},\]

\[\mathcal{M}_{\ell m}[H] = -\sqrt{m+1}\,(\ell+1) H_{\ell, m+1} -\sqrt{m+1}\,\ell H_{\ell-1, m+1} +\sqrt{m}\,\ell H_{\ell, m-1} +\sqrt{m}\,(\ell+1) H_{\ell+1, m-1}.\]

The diamagnetic drive term \(\mathcal{D}_{\ell m}\) follows a Laguerre formulation with explicit \(a/L_n\) and \(a/L_T\) dependence, including a separate coupling in \(m=2\) for temperature-gradient drive.

Streaming acts on \(H\): the parallel derivative is applied to \(H_{\ell m}\) (field terms included at \(m = 0, 1\)) before the Hermite ladder, periodically or along the linked twist-and-shift chains (gkx.terms.linear_terms.streaming_contribution).

Nonlinear E×B and flutter terms

The nonlinear gyrokinetic equation adds the \(E\times B\) bracket and the electromagnetic flutter coupling. In GKX the nonlinear contribution is

\[\left(\frac{\partial g}{\partial t}\right)_\mathrm{NL} = -\left\{ \langle \chi \rangle, g \right\} - v_{th}\,\left(\sqrt{m+1}\,\{\langle A_\parallel \rangle, g\}_{m+1} + \sqrt{m}\,\{\langle A_\parallel \rangle, g\}_{m-1}\right),\]

with the Poisson bracket

\[\{g, \chi\} = \frac{\partial g}{\partial x}\frac{\partial \chi}{\partial y} - \frac{\partial g}{\partial y}\frac{\partial \chi}{\partial x}.\]

The gyrokinetic potential includes the perpendicular magnetic perturbation through

\[\chi = J_\ell \phi + J_\ell^B B_\parallel,\]

so the nonlinear operator naturally splits into \(E\times B\), \(B_\parallel\), and flutter contributions. The implementation in gkx.terms.nonlinear supports standard gyrokinetic normalization: gradients are computed with FFTs in \(x,y\), the bracket is evaluated in real space, and the result is filtered by the de-alias mask before returning to spectral space.