W.N.G. Hitchon

Effect of the ambipolar potential on stellarator confinement

The confinement properties of reactor-scale stellarators in the presence of a self-consistent ambipolar potential Φ are examined in the light of the current theoretical expectations for the dependence of the transport coefficients on collisionality and radial electric field. It is found that stellarators have sufficiently good confinement to be viable as reactors. In addition, it is found that multiple roots of the ambipolarity constraint can exist in stellarators (as in other devices) and that, for appropriate parameters, confinement can be improved by operating at a root different from the one usually considered.

An efficient scheme for convection-dominated transport

Partial differential equations describing transport processes involving a significant effect of the flow velocity may be solved efficiently and easily, using a simple algorithm. The algorithm is based on the propagator(s) (or Greens functions) for the equations of transport theory. The numerical method employed is always at least as fast as finite differencing, and it reduces to a finite difference method in the short time-step limit, but is especially efficient in cases where flow dominates over diffusion and is consequently widely applicable in kinetic theory and fluid dynamics. Using this method, the ion distribution function and the potential in a plasma sheath were calculated in the presence of charge exchange collisions for a wide range of neutral densities.

Electron heating by sheaths in radio frequency discharges

The detailed interaction of individual electrons with oscillating radio frequency (rf) sheaths is considered in this reexamination of the sheath heating process. We develop an analytic expression for the energy delivered to the discharge through sheath heating and that lost due to electron escape to the electrode. Using a simple model of the time‐dependent sheath electric field, we find that electron energy gain due to this process, averaged over an rf period is roughly proportional to the square of the maximum sheath speed and drops slightly with increasing electron temperature. Energy loss due to escaping electrons on the other hand, rises with temperature and ultimately outweighs gain when the average electron velocity exceeds the maximum sheath velocity. The requirement of a net power input to the plasma places upper bounds on bulk electron temperature and density attainable by plasmas to be maintained solely by this mechanism, independent of other aspects of container geometry. However, by increasing...

Solution of master and Fokker-Planck equations by propagator methods, applied to Au/NaCl thin film nucleation

Abstract Novel numerical solutions of Master and Fokker-Planck equations are described and compared for equivalent discrete and continuous problems. The two methods involve the calculation of long-time-step propagator matrices, whose single application is equivalent to many iterations of a finite difference scheme. For the discrete method we present two analytic propagators which are exact for growth-only (no decay) processes, and two approximate propagators for growth and decay processes. The continuous method couples a discrete boundary condition for small clusters with an efficient continuous description for large clusters. These two methods are applied to the nucleation and growth of vapor-deposited thin films whose atoms cluster together to form islands (Volmer-Weber growth). Mobility coalescence of islands is included to show how “slow” nonlinear processes may be included in the model.

Radiation trapping simulations using the propagator function method: complete and partial frequency redistribution

An integral method for solving the problem of imprisonment of resonance radiation based on propagator functions is further developed. Earlier work was restricted to plane parallel and spherical geometries, to a Lorentz lineshape, and to the approximation of complete frequency redistribution. This work extends the method to cylindrical geometry, to a Voigt lineshape, and to include the effects of partial frequency redistribution. The method is ideal for calculating both the time-dependent and the steady-state densities of resonance atoms which result from an arbitrary production rate per unit volume. An emission spectrum is also generated in calculations involving partial frequency redistribution. The propagator function method is at least 50 times faster than the Monte Carlo method. The greater speed of the propagator function method makes it well suited to fully self-consistent kinetic simulations of glow discharge plasmas.

A bounce-averaged Fokker-Planck code for stellarator transport

A computer code for solving the bounce-averaged Fokker-Planck equation appropriate to stellarator transport has been developed and its first applications have been made. The code is much faster than the bounce-averaged Monte-Carlo codes, which up to now have provided the most efficient numerical means for studying stellarator transport. Moreover, since the connection to analytic kinetic theory is more direct for the Fokker-Planck approach than for the Monte-Carlo approach, a comparison of theory and numerical experiment is now possible at a considerably more detailed level than previously.

Physical and numerical verification of discharge calculations

Calculations of radio frequency discharge parameters, and to a lesser extent DC discharge parameters, are apparently highly sensitive to the physical model used. The testing of numerical schemes for error, considering alternative formulations and simple physical models, is carefully considered. Within kinetic models a number of options exist, having different capabilities, the implications of which are examined. Particle simulations are discussed, and mesh-based kinetic calculations are considered in detail. An efficient and accurate mesh-based kinetic model which closely replicates the physical processes taking place is presented. Ways to improve its accuracy, which is limited by the resolution of the mesh, and their effects are presented. Cross-checking of various numerical formulations shows that the results for each are essentially the same. Physical reasoning and simple estimates of discharge parameters are used to further substantiate the predictions for a particular discharge, and the processes taking place in an RF discharge in helium are described in detail. >

Arbitrarily high order Convected Scheme solution of the Vlasov–Poisson system

Abstract The Convected Scheme (CS) is a ‘forward-trajectory’ semi-Lagrangian method for solution of transport equations, which has been most often applied to the kinetic description of plasmas and rarefied neutral gases. In its simplest form, the CS propagates the solution forward in time by advecting the so-called ‘moving cells’ along their characteristic trajectories, and by remapping them on the mesh at the end of the time step. The CS is conservative, positivity preserving, simple to implement, and it is not subject to time step restriction to maintain stability. Recently (Guclu and Hitchon, 2012 [1] ) a new methodology was introduced for reducing numerical diffusion, based on a modified equation analysis: the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping. While the spatial accuracy was increased from 2nd to 4th order, the new scheme retained the important properties of the original method, and was shown to be extremely simple and efficient for constant advection problems. Here the CS is applied to the solution of the Vlasov–Poisson system, which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal Coulomb interactions. The Vlasov equation is split into two constant advection equations, one in configuration space and one in velocity space, and high order time accuracy is achieved by proper composition of the operators. The splitting procedure enables us to use the constant advection solver, which we extend to arbitrarily high order of accuracy in time and space: a new improved procedure is given, which makes the calculation of the corrections straightforward. Focusing on periodic domains, we describe a spectrally accurate scheme based on the fast Fourier transform; the proposed implementation is strictly conservative and positivity preserving. The ability to correctly reproduce the system dynamics, as well as resolving small-scale features in the solution, is shown in classical 1D–1V test cases, both in the linear and the non-linear regimes.

Self-consistent kinetic model of an entire dc discharge

Abstract A dc discharge in helium has been modeled from electrode to electrode, fully kinetically using a self-consistent electric field and including all the physical processes believed to be important for ions, electrons and neutral atoms. The cathode fall, negative glow, and cathode fall-negative glow boundary develop naturally in the simulation without an internal boundary condition and without breaking the electron energy distribution function in components. Results of the simulation are compared to precise experimental results for the electric field profile, species densities, and average electron energy. Excellent agreement is obtained demonstrating that the physical model used is both correct and complete.

Radiation trapping simulations using the propagator function method

Abstract An integral method of solving the Holstein-Biberman equation based on a propagator function is described. This method is used to solve the equation with a Lorentz lineshape in an infinite plane parallel geometry and a hollow spherical geometry. The method is ideal for solving for both the time dependent and the steady state density of resonance atoms which results from an arbitrary production rate per unit volume. The propagator function method is 100 times faster than the Monte Carlo method. The greater speed of the propagator function method makes it well suited to fully self-consistent kinetic simulations of glow discharge plasmas.