Full-F Gyrofluid Model Explained

• Full-F gyrofluid models are formulations that evolve total plasma moments (full densities) instead of small perturbations, capturing nonlinear polarization dynamics.
• They incorporate arbitrary amplitude fluctuations and complete moment hierarchies to simulate complex phenomena such as reconnection and drift-wave turbulence.
• Applications span low-β reconnection, edge turbulence, and gyrokinetic convergence studies, providing a self-consistent framework for nonlinear plasma behavior.

A Full-F gyrofluid model is a gyrofluid formulation in which the total densities, or more generally the full distribution function expanded in moments, are evolved rather than only perturbations about a prescribed background. In the literature surveyed here, this designation includes Hamiltonian full-density reconnection models with nonlocal gyro-Poisson closure, isothermal full-f/full-k turbulence models with arbitrary fluctuation amplitude and arbitrary polarization wavelength, and full-F gyro-moment hierarchies that evolve coefficients of a Hermite–Laguerre expansion of the ion distribution function (Tassi et al., 2011, Kendl, 2023, Frei et al., 2023, Locker et al., 14 Feb 2025). The term is also used carefully in contrast to several reduced gyrofluid systems that are explicitly not Full-F, including minimal electromagnetic closures for Alfvénic turbulence, reduced moment models around homogeneous equilibria, and $\delta f$-type gyrofluid surrogates for plasmoid instability (Bian et al., 2010, Zacharias et al., 2014, Granier et al., 2023, Passot et al., 2024).

1. Definition and distinguishing criteria

In the strongest and most literal usage, a Full-F gyrofluid model evolves full densities or full moments. The Hamiltonian reconnection model of Grasso and collaborators evolves the ion guiding-center density $n_i$, the electron density $n_e$, and the magnetic flux $\psi$, while keeping equilibrium density gradients explicitly as

$n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$

so diamagnetic effects arise self-consistently (Tassi et al., 2011). TIFF defines full-f as arbitrary fluctuation amplitude, so that the total gyrocenter densities $N_s(x,y,t)$ are advanced rather than small perturbations $\tilde N_s$ (Kendl, 2023). The gyro-moment approach for LAPD expands the full distribution function $F_i$ directly, not only a perturbed part, and treats the number of retained moments as an adjustable velocity-space resolution parameter (Frei et al., 2023).

A second defining property is the retention of nonlinear polarization dynamics. In full-f/full-k edge turbulence, quasi-neutrality becomes a generalized variable-coefficient Poisson problem because the polarization coefficient depends on the evolving density field (Kendl, 2023). In GREENY, Full-F is kept in a non-Oberbeck-Boussinesq form, so density variations remain inside the polarization operator rather than being frozen at a constant background (Locker et al., 14 Feb 2025). The 2026 plasmoid study states the same distinction in terms of a non-Oberbeck–Boussinesq arbitrary-wavelength polarization closure and identifies the OB or $\delta F$ model as a reduction obtained by setting $N_z=N_0+\tilde N_z$ and discarding higher-order nonlinearities (Locker et al., 6 Mar 2026).

A recurrent misconception is that any gyrofluid model is automatically Full-F. The record is more specific. The reconnection comparison with EUTERPE states that its gyrofluid model is not a full-F gyrofluid model but a reduced moment model around a fixed equilibrium (Zacharias et al., 2014). The plasmoid-instability model derived from AstroGK is likewise described as a reduced $n_i$0-type gyrofluid model rather than a self-consistent full-$n_i$1 evolution (Granier et al., 2023). The Alfvénic turbulence model of Passot, Sulem, and collaborators is a reduced gyrofluid truncation of gyrokinetics, not a full kinetic model and not a $n_i$2 simulation in the algorithmic sense (Bian et al., 2010). This usage suggests that “Full-F gyrofluid model” is best treated as a specific modeling class, not a generic synonym for gyrofluidity.

2. Dynamical fields, closures, and representative equations

The common structural elements are gyrocenter densities, parallel momenta or parallel flows, electromagnetic potentials, and a gyroaveraged quasi-neutrality closure. In the Hamiltonian low-$n_i$3 reconnection model, the governing equations include

$n_i$4

$n_i$5

closed by

$n_i$6

with Padé gyroaveraging

$n_i$7

The same model introduces the normal fields

$n_i$8

which are advected by stream functions $n_i$9 (Tassi et al., 2011).

GREENY implements a 2D isothermal electromagnetic Full-F gyrofluid model derived from the Full-F equations of Madsen. The evolved fields are total densities and parallel flows for electrons and ions, together with the parallel vector potential $n_e$0. The electron and ion parallel canonical-momentum-like variables,

$n_e$1

are the natural dynamical quantities. Closure is provided by a non-Oberbeck-Boussinesq arbitrary-wavelength polarization equation,

$n_e$2

and by Ampère’s law,

$n_e$3

Ion FLR enters through Padé operators such as

$n_e$4

while electron FLR is neglected because $n_e$5 (Locker et al., 14 Feb 2025).

TIFF uses a different but closely related full-f construction for quasi-two-dimensional edge turbulence. Its basic species equation is

$n_e$6

with

$n_e$7

and a species-dependent effective potential

$n_e$8

Its full-k polarization equation is

$n_e$9

with polarization densities $\psi$0 and $\psi$1 that retain arbitrary-wavelength FLR structure (Kendl, 2023).

The full-F gyro-moment hierarchy extends the same idea into velocity space. It expands the ion distribution as

$\psi$2

where the Hermite basis is shifted by the local ion parallel flow $\psi$3. The lowest coefficients recover fluid moments such as

$\psi$4

This formulation is full-F because it evolves the total distribution through its moments rather than a perturbation hierarchy (Frei et al., 2023).

3. Hamiltonian, variational, and invariant structure

Several Full-F gyrofluid formulations are explicitly Hamiltonian. The diamagnetic reconnection model is formulated as a 2D Hamiltonian electromagnetic gyrofluid model whose equations can be written using a noncanonical Poisson bracket and a conserved Hamiltonian or free-energy functional. In normal-field variables,

$\psi$5

so each transported field is advected by an incompressible flow (Tassi et al., 2011). This Hamiltonian structure is not only formal; in the paper’s interpretation it underlies the nonlinear cascade and reconnection dynamics.

GREENY makes the invariant content explicit through the normalized total energy

$\psi$6

with thermal free energy, perpendicular flow energy, parallel kinetic energy, and magnetic energy contributions. The code also tracks total mass

$\psi$7

and parallel canonical momentum. These are reported to be conserved to reasonable numerical accuracy, up to the effects of numerical dissipation (Locker et al., 14 Feb 2025).

A broader variational setting is provided by the action-principle treatment of generalized fluid motion including gyroviscosity. That work formulates an Eulerian closure principle: the action is built from a Lagrangian flow map but must be expressible entirely in Eulerian observables,

$\psi$8

The resulting Eulerian equations have stress-divergence form,

$\psi$9

and gyroviscous terms appear as nondissipative contributions to the stress tensor (Lingam et al., 2014). That paper explicitly states that its illustrative intrinsic-angular-momentum model is not itself a complete modern Full-F gyrofluid theory, but it supplies a first-principles template for constructing energy- and momentum-conserving gyrofluid or gyroviscous closures.

4. Reduced limits and relation to $n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$0, Oberbeck–Boussinesq, and gyrokinetics

The relation between Full-F and reduced models is one of the central organizing principles of the field. In GREENY, the $n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$1 limit is obtained by setting

$n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$2

and neglecting triple nonlinearities. In that limit the full density factors disappear from advection, Ampère coupling, and polarization, and the closure reduces to the familiar linearized form

$n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$3

The 2026 plasmoid work emphasizes the same point in stronger terms: Full-$n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$4 and Full-$n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$5+OB differ by the presence of cubic nonlinearities, and the OB/$n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$6 approximation is only a reduction, not a different physics model (Locker et al., 14 Feb 2025, Locker et al., 6 Mar 2026).

TIFF provides a complementary hierarchy of limits. It states explicitly that the model reduces to the Hasegawa–Wakatani system in the simultaneous limits of small fluctuation amplitudes, cold ions, and homogeneous magnetic field. The same paper organizes the model family as full-f full-k, full-f low-k, delta-f full-k, and delta-f low-k/HW. This clarifies that “full-f” concerns amplitude nonlinearity, whereas “full-k” concerns arbitrary polarization wavelength (Kendl, 2023).

The gyro-moment formulation blurs the boundary between gyrofluid and gyrokinetic descriptions in a different way. With only a few moments retained, it resembles a gyrofluid model; with many moments retained, it approaches gyrokinetic velocity-space resolution. Its closure is the explicit truncation

$n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$7

which can be systematically relaxed by increasing $n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$8 (Frei et al., 2023).

By contrast, multiple widely cited gyrofluid models are not Full-F. The EUTERPE comparison states that its gyrofluid equations are obtained from the first two velocity-space moments of gyrokinetic equations under constant temperatures and homogeneous equilibrium, and are therefore a reduced moment model rather than a Full-F formulation (Zacharias et al., 2014). The AstroGK-based plasmoid model is closed by a quasi-static closure and evolves only $n_{i,eq}(x)=n_{e,eq}(x)=n_0' x,$9 and $N_s(x,y,t)$0, with ions treated as cold and $N_s(x,y,t)$1 (Granier et al., 2023). The two-field Hamiltonian model for space plasmas is a gyrofluid reduction of gyrokinetics; its discussion notes that “full-F” in that context would mean retaining additional moments and more complete electron/ion FLR coupling, not solving a literal full-$N_s(x,y,t)$2 kinetic equation (Passot et al., 2024).

5. Physical regimes and applications

Full-F gyrofluid models have been deployed mainly in low-$N_s(x,y,t)$3, strong-guide-field, collisionless or nearly collisionless settings where finite Larmor radius effects, electron inertia, and nonlinear polarization are central. In diamagnetic reconnection, increasing $N_s(x,y,t)$4 reduces the linear growth rate and can fully stabilize the mode; increasing ion temperature tends to make the magnetic islands propagate in the ion diamagnetic drift direction; and in the nonlinear regime diamagnetic effects reduce the final island width (Tassi et al., 2011). These results arise in a formulation where equilibrium density gradients are retained explicitly rather than linearized away.

GREENY and the later Full-F plasmoid study place this framework in Harris-sheet reconnection. GREENY reports that decreasing $N_s(x,y,t)$5 makes the reconnection region more elongated and Y-shaped, whereas larger $N_s(x,y,t)$6 gives a more X-shaped geometry; warm ions can generate plasmoid-like structures due to FLR-induced ion–electron decoupling; and excessive hyperviscosity can suppress reconnection altogether (Locker et al., 14 Feb 2025). The 2026 study further reports fast collisionless reconnection rates

$N_s(x,y,t)$7

finds that the linearized operator is strongly non-normal with condition numbers as large as $N_s(x,y,t)$8, and interprets extended pseudospectra and transient amplification as a mechanism for explosive reconnection and the transition from linear tearing to rapid nonlinear acceleration (Locker et al., 6 Mar 2026).

Full-F formulations are equally motivated by edge, scrape-off-layer, and open-field-line turbulence. TIFF targets quasi-two-dimensional interchange and drift-wave turbulence with large-amplitude blobs and filaments, strong radial profile variation, and finite-$N_s(x,y,t)$9 polarization, precisely the regime where small-amplitude and long-wavelength assumptions are expected to fail (Kendl, 2023). In LAPD-oriented full-F gyro-moment simulations, collisions damp higher-order Hermite moments in the high-collisional regime, $\tilde N_s$0 is already sufficient in that regime for the studied setup, and the resulting turbulence is in qualitative agreement with two-fluid Braginskii profiles and statistics (Frei et al., 2023). The later full-f DK/GK hybrid study reaches a similar conclusion for LAPD parameters: the DK ion distribution is approximately bi-Maxwellian, the simulations are dominated by Kelvin–Helmholtz-driven turbulence, and the GK sector becomes dynamically important only when collisionality is reduced and the GK source is amplified (Mencke et al., 13 Mar 2026).

6. Numerical realization, benchmarking, and extensions

The numerical difficulty of Full-F gyrofluid modeling lies chiefly in the inversion of nonlinear field equations. TIFF identifies the generalized elliptic problem

$\tilde N_s$1

as the computational bottleneck and implements three solvers: PCG, SOR, and the dynamically corrected Fourier method. The DCF method builds on Teague’s variable-coefficient Poisson strategy, uses the previous time step to estimate the correction from the Helmholtz decomposition, and is reported to achieve accuracy comparable to the fourth-order PCG solver while being significantly faster in the tests (Kendl, 2023).

GREENY uses Karniadakis time stepping, initial backward Euler seeding, finite differences for the dynamical equations, and Arakawa discretization for Poisson brackets. Its field solvers include Teague’s method, PCG, SOR, and FFT-based inversion. The paper reports solver verification against analytic test problems with approximately $\tilde N_s$2 scaling for SOR and approximately $\tilde N_s$3 scaling for FFT or spectral solvers, and the same $\tilde N_s$4 versus $\tilde N_s$5 split for polarization solves using SOR versus spectral Teague and PCG (Locker et al., 14 Feb 2025). The later plasmoid study stresses that insufficient resolution can suppress plasmoid formation even when nominal instability is high, and reports comparable current structures from an Arakawa discretization with weak hyperdiffusion and from an upwind scheme (Locker et al., 6 Mar 2026).

Moment-based full-F models introduce a second convergence problem in velocity space. In the LAPD gyro-moment study, the collision operator damps high-order moments, $\tilde N_s$6 is generally enough when FLR effects are neglected, and the shifted Hermite basis centered at $\tilde N_s$7 is described as essential for avoiding poor convergence near sonic sheath flows (Frei et al., 2023). This hierarchy-based strategy provides a systematic route from few-moment gyrofluid closures toward higher-fidelity kinetic resolution.

Current extensions move in two directions. One direction is toward broader physics: the four-field Hamiltonian gyrofluid model for space plasmas restores ion density and ion parallel velocity, improves electron FLR fidelity at small or moderate $\tilde N_s$8, and retains coupling to slow magnetosonic waves (Passot et al., 2024). The other direction is toward broader geometry and closure content: TIFF is presented as a reference case for future three-dimensional full-f full-k models, the LAPD gyro-moment framework identifies FLR effects, better sheath conditions, and kinetic electrons as natural next steps, and the action-principle program points to two-fluid gyroviscous actions, anisotropic pressure, and kinetic or gyrokinetic extensions (Kendl, 2023, Frei et al., 2023, Lingam et al., 2014).

In this literature, the Full-F gyrofluid model is therefore not a single canonical set of equations but a modeling principle: evolve total densities or total moments, retain gyroaveraged FLR structure in the closures, and solve polarization and electromagnetic constraints self-consistently even when the coefficients depend on the evolving plasma state. The principal distinction from reduced gyrofluid and $\tilde N_s$9 models is not merely semantic. It lies in the retention of background evolution, density-dependent polarization, and higher-order nonlinearities that become significant in collisionless reconnection, plasmoid formation, and large-amplitude edge or open-field-line turbulence.