Thermodynamic bounds on gyrokinetic instabilities and turbulence

For over half a century, an enormous effort has been devoted to the study of microinstabilities in magnetically confined plasmas through gyrokinetic theory. Thousands of papers and millions of lines of code have been devoted to this subject. Most of the results are of highly specialized nature, e.g. the study of particular modes (ITG, ETG, TEM, KBM, RBM, MTM,…) in specific magnetic geometries, but little is known in general about gyrokinetic microinstaiblities, despite the great effort devoted to their study.

This talk will describe how rigorous upper bounds can be derived on the growth of any electrostatic or electromagnetic instability in gyrokinetics, regardless of the magnetic geometry, number of particle species, plasma pressure, and collisions. These bounds are valid not only for linear instability growth, but also for nonlinear turbulence. The reason why they are so general is that they result from thermodynamic considerations of energy and entropy balance. They reflect the fastest possible instantaneous growth of gyrokinetic fluctuation energy, which is much easier to calculate than conventional eigenmodes. In fact, the calculation can be completed, once and for all, for all gyrokinetic instabilities by solving a low-dimensional matrix eigenvalue problem. Comparisons with numerical simulations will be shown, indicating that the upper bounds are usually close to the maximum growth rate that can be achieved in a theta pinch, which, in turn, is a few times larger than that in a typical tokamak or stellarator. Methods of deriving less general but increasingly stringent bounds will also be described, the overall strategy being to study gyrokinetics by proceeding from general properties to more specific ones rather than the other way around.