The Boltzmann equation introduced in Chapter 2 describes the time evolution of a single particle species distribution. Ostensibly, one could simulate the evolution of Equation 2.8 by approximating the second and third terms (i.e., the phase-space derivatives) at each time step, prescribing a numerical form of the right-hand-side collisional term, and using an appropriate time-stepping scheme to evolve the parti-cle distribution. A popular alternative to following individual partiparti-cles is to follow
“macro-particles” that each represent millions of physical particles. The macro-particles have a prescribed shape that determines their charge density, which the model weights to a grid at each time step. The model can then derive a self-consistent electric field value at each grid point at each time step simply by knowing each parti-cle’s position. The complexity of this approach increases in proportion to the number of particles, N ; for contrast, the complexity of calculating the electric field by eval-uating Coulomb’s Law for each pair of particles would increase in proportion to N2. The method of calculating field quantities by extrapolating particle positions to a grid is called the particle-in-cell (PIC) method (Birdsall and Langdon, 1991).
The Electrostatic Parallel PIC (EPPIC) code, designed by Professor Meers Op-penheim, employs a PIC approach with additional collision routines to simulate the sort of weakly ionized plasma found in the Earth’s E-region ionosphere. Developing a hybrid extension of EPPIC, which treats electrons as an inertialess, isothermal
fluid in quasi-neutrality with ions, has been a major component of this dissertation research. A description of EPPIC, with an emphasis on the hybrid extension, follows.
EPPIC begins by initializing the particles based on a given distribution. A nat-ural initial distribution is to have spatially homogeneous particle positions and a Gaussian distribution of velocities. Such a simple initial condition can nonetheless address many fundamental aspects of kinetic plasma physics. The simulations pre-sented in Chapter 6 initialized particles homogeneously with special modifications designed to reduce the noise inherent in approximating a continuous distribution on a discrete grid. Should the physical model require a more complex initial distribution, the user must chose an algorithm that defines particle positions and velocities in a physically realistic way. One method for defining particle positions is the “transfor-mation method”. The transfor“transfor-mation method assigns particle positions based on the inverse of the integral of the target distribution. It is efficient but it clearly requires that the distribution have a finite, invertible integral. The simulations presented in Chapter 4 initialized particles via the transformation method. Another method for defining particle positions is called the “rejection method”. The rejection method is not specific to the PIC method nor even to plasma models – it is a general technique for generating random deviates from a known distribution function (Press et al., 2007). The simulations presented in Chapter 5 initialized particles via the rejection method.
Following the particle-initialization stage, as well as other setup tasks, EPPIC proceeds to “gather” the particle positions into either quasi-neutral density, n(r), and ion flux, Γi(r) = n(r)vi(r), or into charge density, ρc(x, y, z) = P
jqjnj(r), where the qj are the species’ charges and the nj are the individual species’ densities.
In either case, r represents whatever positional coordinates are appropriate for mod-eling the system of interest. The choice of n and Γi or ρc depends on whether the
physical model assumes quasi-neutrality or not. Given gathered quantities, the gen-eral PIC method would calculate the electric field, E(r), and magnetic field, B(r), on the grid. For a fully electromagnetic system, it would use a gathered current density, J(r), to derive E(r) and B(r) from Maxwell’s equations. However, the simulations presented in this dissertation assume the plasma is electrostatic. Under the electro-static assumption, magnetic perturbations are negligible and Faraday’s Law reads
∇ × E = 0. General rules of vector calculus state that the curl of the gradient of a scalar function is identically zero, meaning that the electrostatic statement of Fara-day’s Law implies E =−∇φ; here, φ(r) is the electrostatic potential and the negative sign ensures that ions and electrons move in the correct directions. The pure-PIC version of EPPIC does not assume quasi-neutrality, and can therefore use Gauss’s Law in the form of Poisson’s equation, ∇2φ = −ρc/0, to calculate the potential.
The scientific computing community has a wealth of experience solving Poisson’s equation; the solution is particularly simple for simulations with periodic boundary conditions. The results presented in Chapter 6 employed a fast spectral method for solving Poisson’s equation with periodic boundary conditions. Quasi-neutrality, on the other hand, assumes by definition that ρc ≈ 0. Such a model cannot take ad-vantage of the the nice properties inherent in Poisson’s equation and must calculate the potential by an alternative approach. The results presented in Chapters 4 and 5 employed the quasi-neutral hybrid version of EPPIC, which solves Equation 2.14 for the potential via a finite difference method.
EPPIC uses a particle parallelization scheme in which multiple processors essen-tially run independent PIC simulations that come together to compute the electric field. Each processor updates particle velocities and positions independently, using the existing electric field, then gathers the positions as described above. Next, EP-PIC uses the Message Passing Interface (MPI) to calculate an average density (and
ion flux, if applicable) across processors, which it passes to the field-solver routine.
This parallelization scheme allows EPPIC to run efficiently on up thousands to tens of thousands of processors.