ORB5X represents each kinetic species by the gyrocenter distribution \(f_s(\mathbf R,v_\parallel,\mu;t)\), where \(\mathbf R\) is the gyrocenter position, \(v_\parallel\) is parallel velocity, and \(\mu\) is magnetic moment. In the principal toroidal coordinates, marker positions are stored as radial coordinate \(s\), straight-field-line poloidal angle \(\chi\), and toroidal angle \(\varphi\). Near the magnetic axis, nonsingular Cartesian-like poloidal coordinates are used by the particle mover.

Variational formulation

The inherited ORB5 model is derived from a gyrokinetic action. This provides a consistent gyrokinetic Vlasov equation and weak forms of the quasineutrality and Ampère equations suitable for finite elements. Electromagnetic simulations can split the parallel vector potential into Hamiltonian and symplectic components,

\[A_\parallel=A_\parallel^{(h)}+A_\parallel^{(s)}, \]

and use the pullback formulation to mitigate the cancellation problem. The available approximation is selected from the run input; not every combination of species, equilibrium, field equation, collision model, and source should be assumed validated.

Low-noise distribution split

The code uses the control-variate decomposition

\[f_s = f_{0s} + \delta f_s, \]

with markers sampling a phase-space density \(g_s\). Marker weights represent the background and perturbation through \(f_0=P g\) and \(\delta f=W g\). For a collisionless marker \(i\), ORB5 evolves the orbit and the selected weight formulation; the direct- \(\delta f\) option exploits invariance of the total distribution along nonlinear collisionless trajectories.

Field equations

Electrostatic potential \(\phi\) and parallel vector-potential components are represented on a tensor-product B-spline basis. Marker moments supply the right-hand sides of the weak quasineutrality and Ampère systems. Depending on the model selection, terms can include gyroaveraged charge density, polarization, adiabatic response, and parallel current.

Equilibria and profiles

orb5x::Equil supplies magnetic geometry, coordinate transforms, safety factor, Jacobian, and equilibrium field quantities. orb5x::BackgroundModule and profile helpers evaluate density, temperature, and distribution functions. These values are normalized internally; consult the definitions in orb5x::BasicVariables, orb5x::EquilVariables, and orb5x::SpeciesVariables before interpreting raw arrays.

Optional physics

The implementation includes numerical antennas, magnetic islands, collisions, heat and particle sources, profile relaxation, noise control, phase-space zonal structures, and three-dimensional diagnostics. Their activation and compatibility are input-controlled. The corresponding module header is the authoritative API description; the adjacent Fortran module remains a parity reference.