Implementation of the Quiet Implicit PIC (QIP) moment equations in toroidal geometry

The Quiet Implicit PIC (QIP) method separates the distribution function into a bulk distribution f0, describing the fluid-like behavior, and a remaining part δf, describing the intrinsically kinetic behavior. For fusion plasmas, f0 is chosen as a local Maxwellian distribution. The evolution of f0 is governed by Maxwell's equations and a set of moment equations which include closure terms depending on δf. The evolution of δf is represented by marker particles which follow the characteristics of the kinetic equation in the PIC (Particle In Cell) technique. The attractiveness of the QIP equations is the potential to efficiently and accurately model the fluid-like behavior of low-collisionality plasmas while retaining kinetic effects through δf. Shot noise is sufficiently low that physically important fluctuations are resolved. The numerical closure of the moment equations provided by QIP also provides more insight into analytical closure schemes. The implementation of QIP divides into two parts. First, the moment equations coupled to Maxwell's equations are solved on a computational grid. Second, particle equations of motion determine δf which is then used to close the moment equations. Disparate time scales of the plasma require implicit time differencing for the moment equation solution. This leads to a large, stiff linear system, whose solution presents the major numerical challenge. Implementation of the first part of the QIP algorithm has recently been completed in the TPCN code by the addition of the pressure evolution equation. Additionally, TPCN now solves the implicitly time-differenced moment-Maxwell equations in a numerically generated orthogonal equilibrium flux coordinate system using the orthogonal trajectory method. This greatly facilitates separating the disparate scales associated with modes parallel and perpendicular to the equilibrium magnetic field