Linear Model And Derivations

This page records the exact linear model currently benchmarked in SPECTRAX-GK. 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 \(k_\perp\)

SPECTRAX-GK 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.