This paper presents a complete theoretical framework for studying turbulence and transport in rapidly rotating tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio ε = ρi/α of the gyroradius to the equilibrium scale length. Proceeding order by order in this expansion, a set of coupled multiscale equations is developed. They describe an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the interplay between the equilibrium and the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equilibrium is obtained from the generalized Grad-Shafranov equation for a rotating plasma, determining the magnetic flux function from the mean pressure and velocity profiles of the plasma. The slow (resistive-timescale) evolution of the magnetic field is given by an evolution equation for the safety factor q. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the 'high-flow' gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local (in space) cascade of the free energy of the fluctuations (i.e. there is no turbulence spreading). Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical and fluctuating corrections to the equilibrium Maxwellian act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived from the transport equations. Total energy, thermal, kinetic and magnetic, is conserved and there is no net turbulent heating. Entropy is produced by the action of fluxes flattening gradients, Ohmic heating and the equilibration of interspecies temperature differences. This equilibration is found to include both turbulent and collisional contributions. Finally, this framework is condensed, in the low-Mach-number limit, to a more concise set of equations suitable for numerical implementation.

Scaling laws for ion temperature gradient driven turbulence in magnetized toroidal plasmas are derived and compared with direct numerical simulations. Predicted dependences of turbulence fluctuation amplitudes, spatial scales, and resulting heat fluxes on temperature gradient and magnetic field line pitch are found to agree with numerical results in both the driving and inertial ranges. Evidence is provided to support the critical balance conjecture that parallel streaming and nonlinear perpendicular decorrelation times are comparable at all spatial scales, leading to a scaling relationship between parallel and perpendicular spatial scales. This indicates that even strongly magnetized plasma turbulence is intrinsically three-dimensional.Introduction. Microscale turbulence is a ubiquitous feature of the plasmas used for magnetic confinement fusion. It is driven by kinetic instabilities feeding predominantly off a strong mean gradient in the ion temperature, and it is responsible for the majority of particle and heat transport observed in experiment. As with neutral fluid and magnetohydrodynamic turbulence, exact analytical results for kinetic plasma turbulence are rare, and numerical simulations are costly. Phenomenological scaling laws are thus useful for guiding simulation and providing gross predictions of plasma behavior in a multi-dimensional parameter space.Experimental, numerical, and analytical results (cf. [1-4]) have long been used to predict the dependence of turbulent fluxes on the mean plasma gradients and on the magnetic field configuration. However, scalings based on empirical observations provide limited physical insight, and the theoretical predictions, which are predominantly based on linear or quasilinear arguments, are not sufficiently detailed to be easily falsifiable. A more detailed examination of the properties of kinetic plasma turbulence has been conducted for scales smaller than the ion Larmor radius [5][6][7], but it is the ion temperature gradient (ITG) driven turbulence above the Larmor scale that is most relevant for heat transport in fusion devices (cf. [8]). Recent advances in plasma fluctuation measurements [9, 10] have provided turbulence spectra in this scale range; direct numerical simulations have also calculated spectra [11] and studied energy injection, transfer, and dissipation [12,13].In this Letter, we propose a phenomenological scaling theory of ITG turbulence. A number of simple, physically-motivated conjectures about the nature of this turbulence are formulated and applied to obtain fluctuation spectra from the driving scale to the ion Larmor scale. We then derive predictions for the dependence of heat flux on plasma current and ion temperature gradient. Numerical results are presented to support our predictions and justify our conjectures.Gyrokinetic turbulence. Plasma fluctuations in a strong mean magnetic field are anisotropic with respect to the mean field direction and have typical frequencies