152 research outputs found
Intrinsic rotation in tokamaks: theory
Self-consistent equations for intrinsic rotation in tokamaks with small
poloidal magnetic field compared to the total magnetic field 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,
, and turbulence with characteristic lengths of the order of the
poloidal gyroradius, . 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
, 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 . 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
and a Poisson equation accurate to order . 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 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 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 in general magnetic geometry. Here
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
. 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 . 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
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