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This paper develops a clear procedure for solving the nonlinear Vlasov–Maxwell equations for a one-component intense charged particle beam or finite-length charge bunch propagating through a cylindrical conducting pipe (radius r=rw=const), and confined by an applied focusing force Ffoc. In particular, the nonlinear Vlasov–Maxwell equations are Lorentz transformed to the beam frame (“primed” variables) moving with axial velocity Vb=βbc=const relative to the laboratory. In the beam frame, the particle motions are nonrelativistic for the applications of practical interest, already a major simplification. Then, in the beam frame, the electrostatic approximation (Es′=−∇′φ′, ET′≃0≃Bs′) is made, which fully incorporates beam space-charge effects, but neglects any fast electromagnetic processes with transverse polarization (e.g., light waves). The resulting Vlasov–Maxwell equations are then Lorentz transformed back to the laboratory frame, and properties of the self-generated fields and resulting nonlinear Vlasov...

We present a novel mechanism for producing an equilibrium potential well near the edge of a tokamak. Briefly, because of the difference in gyroradii between electrons and ions, an equilibrium electrostatic potential is generated in the presence of spatial inhomogeneity of the background plasma, which, in turn, produces a well associated with the radial electric field, Er, as observed at the edge of many tokamak experiments. We will show that this theoretically predicted Er field, which can be regarded as producing a long radial wavelength zonal flow, agrees well with recent experimental measurements. The relationship between the equilibrium configuration used in this study and that of the Woltjer-Taylor state will be discussed.

This paper makes use of the nonlinear Vlasov–Poisson equations to describe the propagation of an intense, non-neutral ion beam through a periodic focusing solenoidal field with coupling coefficient κz(s+S)=κz(s) in the thin-beam approximation (rb≪S). The nonlinear δF formalism is developed for numerical simulation applications by dividing the total distribution function Fb into a zero-order part (Fb0) that propagates through the average focusing field κz=const, plus a perturbation (δFb) which evolves nonlinearly in the zero-order and perturbed field configurations. To illustrate the application of the technique to axisymmetric, matched-beam propagation, nonlinear δF-simulation results are presented for the case where Fb0 corresponds to a thermal equilibrium distribution, and the oscillatory component of the coupling coefficient, δκz(s)=κz(s)−κz, turns on adiabatically over many periods S of the focusing lattice. For adiabatic turn-on of δκz(s) over 20–100 lattice periods, the amplitude of the mismatch oscillation is reduced by more than one order of magnitude compared to the case where the field oscillation is turned on suddenly. Quiescent, matched-beam propagation at high beam intensities is demonstrated over several hundred lattice periods.

In this paper we present a detailed formulation and analysis of the rate equations for statistically averaged quantities for an intense non-neutral beam propagating through a periodic solenoidal focusing field Bsol(x) with axial periodicity length S=const. The analysis is based on the nonlinear Vlasov–Maxwell equations in the electrostatic approximation, assuming a thin beam with characteristic beam radius rb≪S, and small transverse momentum and axial momentum spread in comparison with the directed axial momentum pz=γbmβbc. The global rate equation is derived for the self-consistent nonlinear evolution of the statistical average 〈χ〉=Nb−1∫dXdYdX′dY′χFb, where χ(X,Y,X′,Y′,s) is a general phase function, and Fb(X,Y,X′,Y′,s) is the distribution function of the beam particles in the transverse phase space (X,Y,X′,Y′) appropriate to the Larmor frame. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular momentum 〈Pθ〉, center-of-mass motion for 〈X〉 and ...

Newly-developed perturbative (δf) particle simulation techniques are used to investigate the detailed stability and transport properties of an intense ion beam propagating through an alternating-gradient focusing lattice with initial Kapchinskij-Vladimirskij (KV) distribution. Stability properties are investigated over a wide range of beam current and focusing field strength. In the unstable region, large-amplitude density perturbations with low azimuthal harmonic numbers, concentrated near the beam surface, are observed, and their nonlinear consequences are discussed. In the stable region, it is shown that the KV beam distribution propagates quiescently over large distances (up to 500 lattice periods in the simulations). The δf-scheme used in the present numerical studies exhibits minimal noise and accuracy problems in comparison with standard particle-in-cell simulations.

Internal disruption in a tokamak has been simulated using a three‐dimensional magneto‐inductive gyrokinetic particle code. The code operates in both the standard gyrokinetic mode (total‐f code) and the fully nonlinear characteristic mode (δf code). The latter is a quiet low noise algorithm. The computational model represents a straight tokamak with periodic boundary conditions in the toroidal direction and a square cross section with perfectly conducting walls in the poloidal direction. The linear mode structure of an unstable m=1 (poloidal) and n=1 (toroidal) kinetic internal kink mode is clearly observed, especially in the δf code. The width of the current layer around the x‐point, where magnetic reconnection occurs, is found to be close to the collisionless electron skin depth, indicating the importance of electron inertia. Both codes give very similar nonlinear results, in which full reconnection in the Alfven time scale is observed along with the electrostatic potential structures created during this phase. The resulting E×B drift from the potential dominates the nonlinear phase after the full reconnection.

The gyrokinetic approach for arbitrary frequency dynamics in magnetized plasmas is explored, using the gyrocenter-gauge kinetic theory. Contrary to low-frequency gyrokinetics, which views each particle as a rigid charged ring, arbitrary frequency response of a particle is described by a quickly changing Kruskal ring. This approach allows the separation of gyrocenter and gyrophase responses and thus allows for, in many situations, larger time steps for the gyrocenter push than for the gyrophase push. The gyrophase response which determines the shape of Kruskal rings can be described by a Fourier series in gyrophase for some problems, thus allowing control over the cyclotron harmonics at which the plasma responds. A computational algorithm for particle-in-cell simulation based on this concept has been developed. An example of the ion Bernstein wave is used to illustrate its numerical properties, and comparison with a direct Lorentz-force approach is presented.