Monte Carlo Collision Operator Research Articles

Study of slowing down and thermalization of externally injected ion beams in electron cyclotron resonance ion source plasmas

Electron cyclotron resonance ion source based charge breeder is a promising way to produce rare isotope beams with a high charge state, based on capture and thermalisation of the primary ion beam by the ECR plasma. A simulation technique based on a Monte Carlo collision operator is developed to study the capture of externally injected ion beams in ion source plasmas. To illustrate the utility of the method, we have studied the dynamics of Ar1+ ions in oxygen plasma. Evolution of statistically averaged quantities such as the beam size, average directional velocity, and rms velocity is studied for various beam parameters. It is observed that the number of undamped particles and the steady state beam size increase with the increase in beam emittance. The simulation result indicates that the capture and thermalisation of the ion beam are affected by the kinetic energy and emittance of the beam.Electron cyclotron resonance ion source based charge breeder is a promising way to produce rare isotope beams with a high charge state, based on capture and thermalisation of the primary ion beam by the ECR plasma. A simulation technique based on a Monte Carlo collision operator is developed to study the capture of externally injected ion beams in ion source plasmas. To illustrate the utility of the method, we have studied the dynamics of Ar1+ ions in oxygen plasma. Evolution of statistically averaged quantities such as the beam size, average directional velocity, and rms velocity is studied for various beam parameters. It is observed that the number of undamped particles and the steady state beam size increase with the increase in beam emittance. The simulation result indicates that the capture and thermalisation of the ion beam are affected by the kinetic energy and emittance of the beam.

Integer lattice gas with Monte Carlo collision operator recovers the lattice Boltzmann method with Poisson-distributed fluctuations.

We examine a new kind of lattice gas that closely resembles modern lattice Boltzmann methods. This new kind of lattice gas, which we call a Monte Carlo lattice gas, has interesting properties that shed light on the origin of the multirelaxation time collision operator, and it derives the equilibrium distribution for an entropic lattice Boltzmann. Furthermore these lattice gas methods have Galilean invariant fluctuations given by a Poisson statistics, giving further insight into the properties that we should expect for fluctuating lattice Boltzmann methods.

Energy and momentum preserving Coulomb collision model for kinetic Monte Carlo simulations of plasma steady states in toroidal fusion devices

A kinetic Monte Carlo model suited for self-consistent transport studies is proposed and tested. The Monte Carlo collision operator is based on a widely used model of Coulomb scattering by a drifting Maxwellian and a new algorithm enforcing the momentum and energy conservation laws. The difference to other approaches consists in a specific procedure of calculating the background Maxwellian parameters, which does not require ensemble averaging and, therefore, allows for the use of single-particle algorithms. This possibility is useful in transport balance (steady state) problems with a phenomenological diffusive ansatz for the turbulent transport, because it allows a direct use of variance reduction methods well suited for single particle algorithms. In addition, a method for the self-consistent calculation of the electric field is discussed. Results of testing of the new collision operator using a set of 1D examples, and preliminary results of 2D modelling in realistic tokamak geometry, are presented.

Monte Carlo implementation of a guiding-center Fokker-Planck kinetic equation

A Monte Carlo method for the collisional guiding-center Fokker-Planck kinetic equation is derived in the five-dimensional guiding-center phase space, where the effects of magnetic drifts due to the background magnetic field nonuniformity are included. It is shown that, in the limit of a homogeneous magnetic field, our guiding-center Monte Carlo collision operator reduces to the guiding-center Monte Carlo Coulomb operator previously derived by Xu and Rosenbluth [Phys. Fluids B 3, 627 (1991)]. Applications of the present work will focus on the collisional transport of energetic ions in complex nonuniform magnetized plasmas in the large mean-free-path (collisionless) limit, where magnetic drifts must be retained.

Development of a Nonlinear Collision Operator for GNET Code

A nonlinear collision operator is formulated in the polar coordinate (v, cos θ) in order to study the effect of collisions among high-energy particles on their confinements in toroidal plasmas. The Monte Carlo collision operator is derived to be easily implemented to the orbit following code. The nonlinear collision model is benchmarked with the Boozer collision model for the Maxwellian plasma. This collision model enables the effect of self collisions of α-particles and energetic particles to be investigated on plasma heating, e.g. NBI and ICRF heating.

Understanding the accuracy of Nanbu’s numerical Coulomb collision operator

We investigate the accuracy of and assumptions underlying the numerical binary Monte Carlo collision operator due to Nanbu [K. Nanbu, Phys. Rev. E 55 (1997) 4642]. The numerical experiments that resulted in the parameterization of the collision kernel used in Nanbu’s operator are argued to be an approximate realization of the Coulomb–Lorentz pitch-angle scattering process, for which an analytical solution for the collision kernel is available. It is demonstrated empirically that Nanbu’s collision operator quite accurately recovers the effects of Coulomb–Lorentz pitch-angle collisions, or processes that approximate these (such interspecies Coulomb collisions with very small mass ratio) even for very large values of the collisional time step. An investigation of the analytical solution shows that Nanbu’s parameterized kernel is highly accurate for small values of the normalized collision time step, but loses some of its accuracy for larger values of the time step. Careful numerical and analytical investigations are presented, which show that the time dependence of the relaxation of a temperature anisotropy by Coulomb–Lorentz collisions has a richer structure than previously thought, and is not accurately represented by an exponential decay with a single decay rate. Finally, a practical collision algorithm is proposed that for small-mass-ratio interspecies Coulomb collisions improves on the accuracy of Nanbu’s algorithm.

Monte Carlo collision operators for use with exact trajectory integrators

Classical transport theory in a spherical torus differs from that in a standard tokamak because in the spherical torus the gyroradius has a width comparable to that of a drift orbit, the banana width. In addition, the operating spherical torus experiments have gradient scales that are comparable to the gyroradius of the ions. Transport theories should use the full kinetic equation, which can be solved numerically using Monte Carlo methods. Such a code for the spherical torus could be based on Alan Glasser’s unpublished ORBIT code, which solves the full orbit equations, plus a Monte Carlo equivalent of the collision operator. This paper derives an appropriate Monte Carlo collision operator for use in such a code.

A Monte Carlo model for velocity space effects in low recycling scrape-off layer plasmas

A Monte Carlo model solving only in velocity space is proposed to investigate a low recycling scrape-off layer (SOL) plasma. In the model, the effects of Coulomb collisions are accurately described by a non-linear Monte Carlo collision operator; a conductive heat flux into the SOL is effectively modelled via randomly exchanging the source particles and the SOL particles; secondary electrons are included. A steady state SOL plasma, which satisfies the particle and energy balances and the neutrality constraint, is determined in terms of the total particle and heat fluxes across the separatrix, the edge plasma temperature, the secondary electron emission coefficient and the SOL size. The model gives gross features of the SOL, such as plasma temperatures and densities, the total sheath potential drop and the sheath energy transmission factor. Simulations neglecting unlike collisions are performed. It is found that the potential drop and the electron energy transmission factor in collisional SOL plasmas are in close agreement with theoretical predictions. The present model primarily provides useful information about collisionless SOL plasmas that is difficult to obtain analytically

Vector Implementation of Nonlinear Monte Carlo Coulomb Collisions

A nonlinear Monte Carlo collision model based on Coulomb binary collisions is presented with emphasis on its efficient implementation. The presented Monte Carlo collision operator, which has a simple form, fulfills particle number, momentum, and energy conservation laws quasi-locally and is equivalent to the nonlinear Fokker–Planck operator in the limit of infinitesimally small time interval of the binary collisions. Two vectorizable algorithms are designed for its fast implementation. Drastic speedup is obtained by the algorithms on vector computers. Various test simulations regarding relaxation processes, electrical conductivity, etc. are carried out in the velocity space. The test results, which are in good agreement with theories, and timing results on vector computers show that the present collision model is practically applicable.

Probabilistic approach to Monte Carlo operators

A general method for constructing Monte Carlo collision operators is formulated based on the introduction of a stochastic Liouville equation. The approach is applied to the investigation of a suitable discretized gyrokinetic equation describing the dynamics of a strongly rotating multispecies plasma, in a toroidally axisymmetric configuration. Improved expressions are obtained for the Monte Carlo collision operators for general non-normal (in v space) coordinate systems. As an application, the important case of the bounce-averaged gyrokinetic equation is discussed, in various cases of interest. Finally, using an approximate collision operator, the friction coefficients are evaluated and expressed in terms of energy and momentum restoring coefficients, thus yielding for the Monte Carlo operators a representation convenient for their numerical implementation.

Monte Carlo approach to collisional transport

A linearized discretization approach is proposed for the numerical investigation of the gyrokinetic equation describing the dynamics of a strongly rotating plasma in a toroidally axisymmetric configuration. The discretization scheme allows the numerical evaluation of the neoclassical transport matrix in terms of a suitably discretized distribution function. A basic feature of the discretization scheme here developed is that it permits the introduction of Monte Carlo collision operators to advance in time the gyrokinetic state of a suitable set of test particles. Such operators, which apply to general non-normal gyrokinetic coordinates, are constructed in such a way as to conserve exactly the collisional invariants. A fundamental consequence is that the neoclassical transport coefficients determined in this way fulfill exactly both Onsager’s symmetry relations and the condition of strict ambipolarity for the particle-flux transport coefficients. Such properties are proven to be satisfied independently of the number of test particles used in the discretization scheme.

Construction of Monte Carlo operators in collisional transport theory

A Monte Carlo approach for investigating the dynamics of quiescent collisional magnetoplasmas is presented, based on the discretization of the gyrokinetic equation. The theory applies to a strongly rotating multispecies plasma, in a toroidally axisymmetric configuration. Expressions of the Monte Carlo collision operators are obtained for general v-space nonorthogonal coordinates systems, in terms of approximate solutions of the discretized gyrokinetic equation. Basic features of the Monte Carlo operators are that they fullfill all the required conservation laws, i.e., linear momentum and kinetic energy conservation, and in addition that they take into account correctly also off-diagonal diffusion coefficients. The present operators are thus potentially useful for describing the dynamics of a multispecies toroidal magnetoplasma. In particular, strict ambipolarity of particle fluxes is ensured automatically in the limit of small departures of the unperturbed particle trajectories from some initial axisymmetric toroidal magnetic surfaces.

Neutral beam injection benchmark studies for stellarators/heliotrons

Neutral beam injection in stellarators/heliotrons is studied with Monte Carlo codes that treat the initial beam deposition and the fast-ion thermalization process. The birth deposition model carefully treats the geometry of the vacuum vessel and includes beam divergence, focusing, and aperture losses. The thermalization process is determined by integrating the guiding centre equations of the fast ions and simulating collisions with the plasma by Monte Carlo collision operators. This process may include charge exchange and neutral reabsorption. For the purposes of this benchmark, we review the different formulations of the guiding centre equations and the Monte Carlo collision operators. We studied perpendicular injection into Heliotron-E, which is located at the Plasma Physics Laboratory at Kyoto University. The magnetic fields of Heliotron-E are computed using the Biot-Savart law with realistic filament models. The sensitivity of the computed heating efficiency to the modelling of the particle loss boundary and to the numerical procedures is examined. The results of three different codes were compared. When the codes solve the same problem, the answers agree quite well. However, changing some of the modelling assumptions (such as the loss boundary location) can create significant differences in the results.

A bounce-averaged Monte Carlo collision operator and ripple transport in a tokamak

A bounce-averaged Monte Carlo pitch-angle scattering operator presented by Tolliver [Phys. Fluids 28, 1083 (1985)] is discussed and shown to follow directly from a bounce-averaged drift-kinetic equation of Connor and Cordey [Nucl. Fusion 14, 185 (1974)]. This operator is used in a discrete mapping analysis of radial transport of test particles in a slightly rippled tokamak, illustrating time step restrictions for its validity. This treatment of collisions is different than that of Yushmanov [Nucl. Fusion 23, 1599 (1983)] but gives the same results for ripple transport.

Monte Carlo studies of transport in stellarators

Transport is studied in toroidal geometry by integrating the guiding-center equations in magnetic coordinates and simulating collisions with a Monte Carlo collision operator. The effects of the ambipolar electric field on diffusion losses are determined for model magnetic fields and the correct magnetic field of the Advanced Toroidal Facility (ATF-1) stellarator. Comparisons are made of the computed diffusion coefficients and the theoretically predicted values.

Properties of the Monte Carlo Lorentz collision operator

The validity of the Monte Carlo collision operator for generating solutions of the reduced Fokker–Planck equation governing suprathermal ions and electrons in a target plasma is established by statistical tests.

Charged-Particle Orbits in Sheared Magnetic Fields; Implications to Diffusion

Charged-particle motion across a sheared magnetic field and electrostatic wave is studied. Resonant-particle orbit widths are derived; two parametric dependences are found in the limits that the particle is near or far from the k/sub parallel/=0 plane, with a smooth transition between the two. The orbit widths are used to calculate diffusion coefficients in tokamak-type configurations. These predictions are confirmed by test-particle computer simulations in (x, y, v/subx/, v/suby/, v/subz/) coordinates employing a Monte Carlo collision operator. (AIP)