A second set of ideas from this toolkit relates to multiscale bifurcation analysis. Consider the following highly simplified problem, in which the turbulent heat flux in some system is a function of a single parameter, the temperature gradient, and for which a sequence of turbulence simulations which fully resolve the turbulent dynamics produces the figure above. One expects to find a bifurcation in the corresponding macroscopic system. That is, as one slowly increases the heat flux from near zero, the temperature gradient that one observes rises from a threshold value of R/L_T ~ 2.5 to R/L_T ~ 4.5. Around this point, further increase of the heat flux results in a rapid doubling of the temperature gradient. (There is hysteresis in such a system as well, indicated by the lower arrow.) To find the critical heat flux, beyond which the temperature gradient suddenly increases, one must presently ``design'' the numerical experiment represented above by hand, and then analyze the results for regions of macroscopic instability. Although tedious, this approach is straightforward. However, realistic transport barrier bifurcation scenarios involve several fluxes (electron and ion heat, parallel and perpendicular momentum, particles, etc.) and a large number of control parameters. It is not realistic to expect to find critical points by hand in these circumstances.

Bifurcation analysis of similarly complicated systems is a well-developed subject in engineering and applied math. Sophisticated solvers exist for finding the critical parameters in such systems in the presence of nonlinearity and noise. The central idea of the multiscale bifurcation analysis is to embed the microscale solver — the turbulence code — within existing engineering solvers. Together with the averaging and restart techniques pioneered by Kevrekidis, this approach allows the computer to ``design'' and carry out the numerical experiments directly. We propose to adapt these techniques for plasma physics, with specific application to the problem of directly calculating the key aspects of transport barrier formation.