CMPD scientists are attacking critical multiscale plasma physics problems, including the sawtooth crash, neoclassical island growth and transport barrier dynamics, with techniques recently developed in the engineering and applied math communities. These generic techniques are designed to bridge the enormous gap between the microscopic description and macroscopic descriptions of a complex physical system and have been successfully applied in a variety of settings. The greatest success has been in situations where the physical system manifests a clearly separation of scales. This characterizes the three examples discussed previously in the context of fusion science quite well. The techniques developed range from the most powerful and general ``equation-free'' approach of Yannis Kevrekidis and the projective integration techniques of Bill Gear to the utilization of ``mean-field-equations'' with transport coefficients calculated from embedded kinetic models. The former has been recognized recently by the J. D. Crawford Prize of SIAM. Even the latter, which has previously been an approach of choice in fusion science, can benefit from the ``gaptooth''and ``ensemble'' techniques, which can potentially greatly improve the efficiency of multiscale computations.

An example of what this approach will mean in a fusion context is the following. We know how to integrate the nonlinear gyrokinetic equations in a local flux tube. However, because these simulations are spatially localized and designed to resolve very fast phenomena — Alfvén waves, parallel electron motion, and so on — they cannot be used for a direct numerical simulation of a full scale burning plasma device for long times. Nor can they be easily employed to calculate the conditions under which a bifurcation of the transport fluxes may occur (i.e.,) the problem of transport barrier formation). However in principal the local short time averaged fluxes of heat, particles and momentum calculated from simulations of many flux tubes across a discharge can be used to evolve the profiles of temperature, density and momentum (the ``slow manifold''). In the ``equation-free'' approach this can be done without specifying or calculating the functional form of the fluxes in terms of the local gradients.

Kevrekidis and Gear bring to the Center a toolkit that has been developed to study system behavior at a coarse, "macroscopic" level of observation using models at a different ("fine", detailed) level of description. This "equation-free" toolkit includes coarse projective integration, coarse bifurcation analysis (Newton-Krylov methods using coarse timesteppers) as well as gaptooth and patch dynamics schemes for spatially distributed systems. These techniques are well-matched to plasma physics problems in fusion science: profile evolution under the influence of turbulence, transport barrier formation, the sawtooth crash, and the growth of the neoclassical tearing mode. Kevrekidis has extensive experience with the equation-free, coarse computer-assisted analysis of kinetic Monte Carlo, Brownian Dynamics, Molecular Dynamics and Lattice Boltzmann problems. In the context of the tasks described in this proposal, integration of the kinetic equations with existing kinetic plasma kernels (GS2, p3d, etc.,) will constitute the "fine scale" model realizations, while observations of suitable short-time averages from such codes constitute the macroscopic observable fields. Even if no closed macroscopic equations are explicitly known for these fields, they can be stably advanced in time with macroscopic timesteps.