Generalized Fishbone-Like Dispersion Relation

Updated 17 December 2025

GFLDR is a unified theoretical framework that generalizes classical fishbone instability by incorporating diverse kinetic and MHD effects.
It integrates thermal ion compressibility, diamagnetic drifts, and wave–particle resonances into a set of coupled integro-differential equations.
The framework accurately predicts various Alfvénic modes and instability thresholds validated through experimental observations in tokamaks like DIII-D and HL-2A.

The Generalized Fishbone-Like Dispersion Relation (GFLDR) is a unified theoretical framework for describing the stability and dynamics of low-frequency shear Alfvén branch modes in toroidal magnetized plasmas. It generalizes the classic energetic-particle-driven fishbone instability to encompass a wide spectrum of instabilities, including kinetic beta-induced Alfvén eigenmodes (BAEs), kinetic ballooning modes (KBMs), beta-induced Alfvén-acoustic eigenmodes (BAAEs), and Alfvénic ion-temperature-gradient (AITG) modes. The GFLDR systematically incorporates thermal ion compressibility, diamagnetic drifts, non-adiabatic kinetic responses, magnetic-curvature–drift coupling, continuous spectrum coupling, and wave–particle resonances, providing detailed predictive tools validated against numerical simulations and experimental observations in devices such as DIII-D and HL-2A (Chavdarovski et al., 2022, Chen et al., 2016, Ma et al., 2023, Falessi et al., 2019).

1. Theoretical Structure and General Formulation

The GFLDR is not a closed-form polynomial dispersion relation but a set of coupled integro-differential equations composed of the kinetic vorticity equation and the quasineutrality constraint. In its canonical form, for perturbations near resonant magnetic surfaces in a magnetized plasma, the core structure is

$i\,\Lambda(\omega) = \delta\overline W_{f}(\omega) + \delta\overline W_{k}(\omega)$

where

$i\,\Lambda(\omega)$ is the generalized inertia term, encapsulating ideal MHD inertia, kinetic corrections (trapped and circulating particle kinetics), and core plasma diamagnetic effects,
$\delta\overline W_{f}(\omega)$ is the fluid (ideal-MHD) potential energy,
$\delta\overline W_{k}(\omega)$ aggregates the kinetic contributions of both thermal and energetic particle wave–particle resonances.

The inertia term exhibits the key form

$\frac{\Lambda^2}{I_\Phi} = \frac{\omega^2}{\omega_A^2} \left(1 - \frac{\omega_{*pi}}{\omega} \right) + \Lambda^2_{\mathrm{cir}}(\omega) + \Lambda^2_{\mathrm{tra}}(\omega)$

where $I_\Phi$ encodes non-ideal effects, and $\omega_{*pi}$ is the ion diamagnetic drift frequency.

This construction allows the GFLDR to encode the continuous transformation of instabilities from classical MHD to kinetic and diamagnetic-drift-dominated regimes (Chavdarovski et al., 2022, Chen et al., 2016). In the high-frequency ideal MHD limit, the GFLDR reverts to the classic pressure coupling equation, recovering fishbone and TAE dispersion relations (Wang et al., 2010).

2. Physical Ingredients and Mathematical Components

The GFLDR synthesizes several disparate physical effects:

Thermal ion compressibility: Parallel inertia and Landau damping.
Diamagnetic drifts: Both ion and energetic particle diamagnetic corrections.
Non-adiabatic kinetic responses: Drift-kinetic modifications captured via special functionals (e.g., $F, G, N, S$ ).
Magnetic-curvature–drift coupling: Geodesic acoustic effects.
Continuum coupling: $k_\parallel v_A$ terms linking the mode to the Alfvén continuum.
Wave–particle resonances: Precession, bounce, and transit interactions for both thermal and energetic particle populations.

A schematic summary of the central mathematical entities is given in the table below:

GFLDR Term	Symbol	Physical Origin
Plasma inertia	$i\Lambda(\omega)$	Fluid and kinetic inertia effects
Fluid potential energy	$\delta \overline{W}_f$	MHD (ballooning/interchange drive)
Kinetic resonant energy	$\delta \overline{W}_k$	Particle resonances (Landau, precession)

Fluid energy coefficients depend on magnetic shear $s$ , pressure gradient parameters $\alpha$ , and the local safety factor $q$ profile (Ma et al., 2023). Kinetic terms are evaluated by integrals over particle distributions, including resonant denominators for precession and bounce resonance (Chavdarovski et al., 2022).

3. Regimes and Branch Structure of Alfvénic Instabilities

The GFLDR provides a unified interpolation between multiple branches of Alfvénic modes:

Kinetic Beta-Induced Alfvén Eigenmode (BAE): $\omega_*^{pi} \ll \sqrt{7/4+\tau}\,q\omega_{ti}$ , with frequency at the acoustic minimum.
Kinetic Ballooning Mode (KBM): $\omega_*^{pi} \gg \sqrt{7/4+\tau}\,q\omega_{ti}$ , with frequencies tracking the ion diamagnetic drift.
Alfvénic Ion-Temperature-Gradient (AITG) mode: Intermediate regime with peak growth near $\omega_*^{pi}/\omega_{ti} \sim \sqrt{7/4+\tau}\,q$ (Chen et al., 2016).
Beta-induced Alfvén-Acoustic Eigenmode (BAAE): Lower-frequency, mixed-polarization roots, especially evident at low $\beta$ and in the presence of strong ion-acoustic coupling (Chavdarovski et al., 2022, Falessi et al., 2019).

The GFLDR master equation

$D(\omega) = \tilde{I}(\omega) - \Delta \widehat{W}_f(\omega) - \Delta \widehat{W}_k(\omega) = 0$

is solved numerically or asymptotically to determine instability thresholds, root frequencies, and growth rates for each mode as dictated by the underlying profile parameters.

4. Experimental Manifestation and Predictive Validity

The GFLDR framework accurately reproduces critical features observed in major tokamak experiments:

In DIII-D, both low-frequency Alfvénic modes (LFAM/KBM) and BAEs demonstrate frequency spectra, mode structures, and radial localization that align with the predictions of local and global GFLDR, including the sharp reactive-type instability peaking at rational $q_\text{min}$ for LFAMs and the broader existence interval for BAEs (Ma et al., 2023).
In HL-2A, GFLDR-E describes the observed AITG activity, demonstrating how weak magnetic shear and moderate pressure gradients can destabilize extended Alfvénic modes, which may be implicated in the formation of internal transport barriers and pedestal regulation (Chen et al., 2016).

Key aspects, such as the sensitivity of LFAM to rational $q_\text{min}$ crossings, the displacement of BAE eigenfunctions to the region of maximum energetic particle drive, and the frequency bands' dependence on the safety factor, are quantitatively matched by theory (Ma et al., 2023).

5. Advanced Applications: Mode Coupling and Multi-Scale Phenomena

The GFLDR supports hybridization (mode-coupling) between different branches through off-diagonal coupling coefficients that scale with the diamagnetic drift. For instance, as the ion diamagnetic frequency increases, the KBM branch acquires an acoustic component and vice versa, with a frequency splitting that becomes pronounced at high $\omega_{*pi}/\omega$ (Chavdarovski et al., 2022).

The formalism allows for the inclusion of both non-resonant and resonant energetic particle effects, enabling predictions of shifting stability thresholds as auxiliary heating or fast-particle populations are varied. Nontrivial radial mode structures (e.g., Hermite-Gaussian eigenfunctions, L=0,1,...) and their localization relative to $q_\text{min}$ and pressure gradient maxima are natural results of the global GFLDR construction (Ma et al., 2023).

6. Computational and Analytical Methodologies

Numerical implementations of the GFLDR employ extended hybrid MHD-gyrokinetic codes (e.g., XHMGC) that incorporate full kinetic response, thermal ion compressibility, and energetic particle dynamics within the GFLDR's analytical framework (Wang et al., 2010). In axisymmetric toroidal geometry, the computation of the continuous spectrum and extraction of the inertia term is facilitated by Floquet and Hill's equation approaches, enabling rigorous mapping of spectrum gaps and polarization properties (i.e., Alfvénicity measure) in experimental equilibria (Falessi et al., 2019).

These numerical techniques clarify the role of ballooning space variables, $\eta$ , periodic eigenfunction structure, and monodromy matrices in the determination of spectrum and global mode stability.

7. Physical Interpretation and Relevance for Tokamak Performance

The GFLDR has profound implications for the operational space and stability of burning plasma devices:

It sets upper bounds on plasma $\beta$ and density peaking by quantifying the onset of ballooning-Alfvénic instabilities.
In weak magnetic shear or reversed-shear configurations, the GFLDR elucidates conditions susceptible to rapid transitions, minor disruptions, or ITB formation.
In the pedestal region of H-mode plasmas, the universal form of the GFLDR predicts the spectral and stability properties that regulate height and width under operational constraints (Chen et al., 2016).

A plausible implication is that future burning plasma scenarios (e.g., ITER) must proactively consider the interconnected GFLDR-predicted instabilities, especially under approaches seeking to avoid neoclassical tearing through low-shear equilibria. Energetic particle drive and core-plasma diamagnetic modifications must be tightly integrated into stability predictions for advanced tokamak regimes.

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