Intrinsic rotation in tokamaks: theory

Self-consistent equations for intrinsic rotation in tokamaks with small poloidal magnetic field $B_p$ compared to the total magnetic field $B$ are derived. The model gives the momentum redistribution due to turbulence, collisional transport and energy injection. Intrinsic rotation is determined by the balance between the momentum redistribution and the turbulent diffusion and convection. Two different turbulence regimes are considered: turbulence with characteristic perpendicular lengths of the order of the ion gyroradius, $\rho_i$, and turbulence with characteristic lengths of the order of the poloidal gyroradius, $(B/B_p) \rho_i$. Intrinsic rotation driven by gyroradius scale turbulence is mainly due to the effect of neoclassical corrections and of finite orbit widths on turbulent momentum transport, whereas for the intrinsic rotation driven by poloidal gyroradius scale turbulence, the slow variation of turbulence characteristics in the radial and poloidal directions and the turbulent particle acceleration can be become as important as the neoclassical and finite orbit width effects. The magnetic drift is shown to be indispensable for the intrinsic rotation driven by the slow variation of turbulence characteristics and the turbulent particle acceleration. The equations are written in a form conducive to implementation in a flux tube code, and the effect of the radial variation of the turbulence is included in a novel way that does not require a global gyrokinetic formalism.Comment: 88 pages, 4 figure

Phase-space Lagrangian derivation of electrostatic gyrokinetics in general geometry

Gyrokinetic theory is based on an asymptotic expansion in the small parameter $\epsilon$, defined as the ratio of the gyroradius and the characteristic length of variation of the magnetic field. In this article, this ordering is strictly implemented to compute the electrostatic gyrokinetic phase-space Lagrangian in general magnetic geometry to order $\epsilon^2$. In particular, a new expression for the complete second-order gyrokinetic Hamiltonian is provided, showing that in a rigorous treatment of gyrokinetic theory magnetic geometry and turbulence cannot be dealt with independently. The new phase-space gyrokinetic Lagrangian gives a Vlasov equation accurate to order $\epsilon^2$ and a Poisson equation accurate to order $\epsilon$. The final expressions are explicit and can be implemented into any simulation without further computations.Comment: 55 pages. Version with typo in equation (135) corrected. The second term in the second line of (135) was missing the subindex that indicates that only the perpendicular component of the gradient enters this ter

Radial penetration of flux surface shaping in tokamaks

Using analytic calculations, the effects of the edge flux surface shape and the toroidal current profile on the penetration of flux surface shaping are investigated in a tokamak. It is shown that the penetration of shaping is determined by the poloidal variation of the poloidal magnetic field on the surface. This fact is used to investigate how different flux surface shapes penetrate from the edge. Then, a technique to separate the effects of magnetic pressure and tension in the Grad-Shafranov equation is presented and used to calculate radial profiles of strong elongation for nearly constant current profiles. Lastly, it is shown that more hollow toroidal current profiles are significantly better at conveying shaping from the edge to the core.Comment: 11 pages, 13 figure

Turbulent momentum pinch of diamagnetic flows in a tokamak

The ion toroidal rotation in a tokamak consists of an $E\times B$ flow due to the radial electric field and a diamagnetic flow due to the radial pressure gradient. The turbulent pinch of toroidal angular momentum due to the Coriolis force studied in previous work is only applicable to the $E\times B$ flow. In this Letter, the momentum pinch for the rotation generated by the radial pressure gradient is calculated and is compared with the Coriolis pinch. This distinction is important for subsonic flows or the flow in the pedestal where the two types of flows are similar in size and opposite in direction. In the edge, the different pinches due to the opposite rotations can result in intrinsic momentum transport that gives significant rotation peaking.Comment: 5 pages and 3 figure

Extension of gyrokinetics to transport time scales

Gyrokinetic simulations have greatly improved our theoretical understanding of turbulent transport in fusion devices. Most gyrokinetic models in use are delta-f simulations in which the slowly varying radial profiles of density and temperature are assumed to be constant for turbulence saturation times, and only the turbulent electromagnetic fluctuations are calculated. New massive simulations are being built to self-consistently determine the radial profiles of density and temperature. However, these new codes have failed to realize that modern gyrokinetic formulations, composed of a gyrokinetic Fokker-Planck equation and a gyrokinetic quasineutrality equation, are only valid for delta-f simulations that do not reach the longer transport time scales necessary to evolve radial profiles. In tokamaks, due to axisymmetry, the evolution of the axisymmetric radial electric field is a challenging problem requiring substantial modifications to gyrokinetic treatments. In this thesis, I study the effect of turbulence on the global electric field and plasma flows. By studying the current conservation equation, or vorticity equation, I prove that the long wavelength, axisymmetric flow must remain neoclassical and I show that the tokamak is intrinsically ambipolar, i.e., the radial current is zero to a very high order for any long wavelength radial electric field. Intrinsic ambipolarity is the origin of the problems with the modern gyrokinetic approach since the lower order gyrokinetic quasineutrality (if properly evaluated) is effectively independent of the radial electric field. I propose a new gyrokinetic formalism to solve for the global radial electric field.Comment: MIT Thesis, 202 pages, 11 figure

Up-down symmetry of the turbulent transport of toroidal angular momentum in tokamaks

Two symmetries of the local nonlinear delta-f gyrokinetic system of equations in tokamaks in the high flow regime are presented. The turbulent transport of toroidal angular momentum changes sign under an up-down reflection of the tokamak and a sign change of both the rotation and the rotation shear. Thus, the turbulent transport of toroidal angular momentum must vanish for up-down symmetric tokamaks in the absence of both rotation and rotation shear. This has important implications for the modeling of spontaneous rotation.Comment: 15 pages, 2 figure

Long-wavelength limit of gyrokinetics in a turbulent tokamak and its intrinsic ambipolarity

Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order $\epsilon^2$ in general magnetic geometry. Here $\epsilon$ is the gyrokinetic expansion parameter, the gyroradius over the macroscopic scale length. Starting from these results, the long-wavelength limit of the gyrokinetic Fokker-Planck and quasineutrality equations is taken for tokamak geometry. Employing the set of equations derived in the present article, it is possible to calculate the long-wavelength components of the distribution functions and of the poloidal electric field to order $\epsilon^2$. These higher-order pieces contain both neoclassical and turbulent contributions, and constitute one of the necessary ingredients (the other is given by the short-wavelength components up to second order) that will eventually enter a complete model for the radial transport of toroidal angular momentum in a tokamak in the low flow ordering. Finally, we provide an explicit and detailed proof that the system consisting of second-order gyrokinetic Fokker-Planck and quasineutrality equations leaves the long-wavelength radial electric field undetermined; that is, the turbulent tokamak is intrinsically ambipolar.Comment: 70 pages. Typos in equations (63), (90), (91), (92) and (129) correcte

Optimized up-down asymmetry to drive fast intrinsic rotation in tokamaks

Breaking the up-down symmetry of the tokamak poloidal cross-section can significantly increase the spontaneous rotation due to turbulent momentum transport. In this work, we optimize the shape of flux surfaces with both tilted elongation and tilted triangularity in order to maximize this drive of intrinsic rotation. Nonlinear gyrokinetic simulations demonstrate that adding optimally-tilted triangularity can double the momentum transport of a tilted elliptical shape. This work indicates that tilting the elongation and triangularity in an ITER-like device can reduce the energy transport and drive intrinsic rotation with an Alfv\'{e}n Mach number on the order of $1\%$. This rotation is four times larger than the rotation expected in ITER and is sufficient to stabilize MHD instabilities. It is shown that this optimal shape can be created using the shaping coils of several experiments.Comment: 16 pages, 5 figure