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Magnetic confinement fusion reactors suffer severely from heat and particlelosses through turbulent transport, which has inspired the construction of everlarger and more expensive reactors. Numerical simulations are vital to theirdesign and operation, but particle collisions are too infrequent for fluiddescriptions to be valid. Instead, strongly magnetised fusion plasmas aredescribed by the gyrokinetic equations, a nonlinear integro-differential systemfor evolving the particle distribution functions in a five-dimensional positionand velocity space, and the consequent electromagnetic field. Due to the highdimensionality, simulations of small reactor sections require hundreds ofthousands of CPU hours on High Performance Computing platforms.  We develop a Hankel-Hermite spectral representation for velocity space thatexploits structural features of the gyrokinetic system. The representationexactly conserves discrete free energy in the absence of explicit dissipation,while our Hermite hypercollision operator captures Landau damping with fewvariables. Calculation of the electromagnetic fields becomes purely local,eliminating inter-processor communication in, and vastly accelerating, searchesfor linear instabilities. We implement these ideas in SpectroGK, an efficientparallel code.  Turbulent fusion plasmas may dissipate free energy through linear phasemixing to fine scales in velocity space, as in Landau damping, or through anonlinear cascade to fine scales in physical space, as in hydrodynamicturbulence. Using SpectroGK to study saturated electrostatic drift-kineticturbulence, we find that the nonlinear cascade suppresses linear phase mixingat energetically-dominant scales, so the turbulence is fluid-like. We use thisobservation to derive Fourier-Hermite spectra for the electrostatic potentialand distribution function, and confirm these spectra with simulations.

Gyrokinetic simulations of fusion plasmas using a spectral velocity space representation