Linear Model And Derivations

This page records the exact linear model currently benchmarked in GKX. It is intended as the technical baseline for the nonlinear extension.

Scope

Current linear validation scope:

  • Cyclone base case (adiabatic electrons)

  • ETG (two-species kinetic ions/electrons, electrostatic limit)

  • KBM beta scan (electromagnetic, \(A_\parallel\) enabled, \(B_\parallel\) disabled)

Normalized equation

For species \(s\), we evolve Laguerre-Hermite moments \(G^{(s)}_{\ell m}(k_x, k_y, z, t)\) in a flux tube:

\[\partial_t G_{\ell m} + \mathcal{S}_{\ell m}[H] + \mathcal{M}_{\ell m}[H] + \mathcal{C}_{\ell m}[H] + \mathcal{G}_{\ell m}[H] = \mathcal{D}_{\ell m}[\phi] + \mathcal{K}_{\ell m}[G],\]

where:

  • \(\mathcal{S}\) is parallel streaming (Hermite ladder)

  • \(\mathcal{M}\) is mirror coupling (\(b'(z)\) geometry factor)

  • \(\mathcal{C}\) and \(\mathcal{G}\) are curvature and grad-\(B\) drifts

  • \(\mathcal{D}\) is diamagnetic drive from \(R/L_n, R/L_T\)

  • \(\mathcal{K}\) contains collisions/hypercollisions/end damping

The field-coupled 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 \(J_\ell^B = J_\ell + J_{\ell-1}\).

Hermite-Laguerre projection

The reduced distribution is expanded as

\[g_s = \sum_{\ell=0}^{N_\ell-1}\sum_{m=0}^{N_m-1} G^{(s)}_{\ell m}\, \mathcal{L}_\ell(\mu)\,\mathcal{H}_m(v_\parallel),\]

with orthogonality

\[\langle \mathcal{H}_m \mathcal{H}_n \rangle = \delta_{mn}, \qquad \langle \mathcal{L}_\ell \mathcal{L}_j \rangle = \delta_{\ell j}.\]

This gives sparse couplings:

  • streaming: \(m \leftrightarrow m\pm1\)

  • curvature: \(m \leftrightarrow m, m\pm2\)

  • grad-\(B\): \(\ell \leftrightarrow \ell, \ell\pm1\)

  • mirror: mixed \((\ell\pm1, m\pm1)\) stencil

These sparsity patterns are implemented as fused tensor kernels in the RHS.

Gyroaverage and \(k_\perp\)

GKX uses the Laguerre gyroaverage

\[J_\ell(b) = \frac{1}{\ell!}\left(-\frac{b}{2}\right)^\ell e^{-b/2}, \qquad b = k_\perp^2 \rho_s^2.\]

For s-alpha geometry, the metric coefficients \(gds2, gds21, gds22\) define

\[k_\perp^2 = \left[ k_y^2 gds2 + 2 k_x k_y \hat{s}^{-1} gds21 + (k_x \hat{s}^{-1})^2 gds22 \right] B^{-2}(z),\]

including linked/NTFT corrections when enabled.

Field equations

Electrostatic limit:

\[\mathcal{Q}\,\phi = \sum_s Z_s n_s \sum_\ell J_\ell G_{\ell 0}^{(s)}.\]

Electromagnetic linear closure:

  • coupled solve for \((\phi, B_\parallel)\) from quasineutrality and perpendicular Ampere

  • \(A_\parallel\) from parallel Ampere

The current KBM benchmark uses \(A_\parallel\) on and \(B_\parallel\) off.

Growth-rate/frequency extraction

Two production paths are used:

  1. Log-linear fit on complex mode signal \(s(t)\):

    \[s(t) \approx s_0 e^{(\gamma - i\omega)t} \Rightarrow \log |s| = \gamma t + c,\quad \arg s = -\omega t + c_\phi.\]

  2. Instantaneous ratio diagnostic (for consistency checks):

    \[r_n = \frac{s_{n+1}}{s_n}, \quad \gamma_n = \frac{\log |r_n|}{\Delta t}, \quad \omega_n = -\frac{\arg(r_n)}{\Delta t}.\]

Windows are selected from intervals with sustained log-linear behavior and finite amplitude support.

Numerical realization

  • perpendicular Fourier representation in \((k_x, k_y)\)

  • field-aligned \(z\) grid with linked boundary support

  • JAX-fused RHS with cache-backed geometry/gyroaverage tensors

  • diffrax and custom fixed-step integrators

  • matrix-free Krylov/shift-invert for eigenvalue-focused scans

Benchmark contract

For all linear benchmark plots/tables:

  • report \(\gamma,\omega\) with explicit normalization

  • publish mismatch tables by scan coordinate (\(k_y\) or \(\beta_{ref}\))

  • include per-case parameter tables (geometry, gradients, species, toggles, grid, resolution)

  • use GX (s-alpha geometry) as the electromagnetic cross-code baseline for KBM

The nonlinear roadmap builds directly on this operator decomposition and normalization contract.