Solution of ordinary differential equations by means of Lie series

Solution of ordinary differential equations by Lie series - Laplace transformation, Weber parabolic-cylinder functions, Helmholtz equations, and applications in physic

Lie series for celestial mechanics, accelerators, satellite stabilization and optimization

Lie series applications to celestial mechanics, accelerators, satellite orbits, and optimizatio

Zonal flows and long-distance correlations during the formation of the edge shear layer in the TJ-II stellarator

A theoretical interpretation is given for the observed long-distance correlations in potential fluctuations in TJ-II. The value of the correlation increases above the critical point of the transition for the emergence of the plasma edge shear flow layer. Mean (i.e. surface averaged, zero-frequency) sheared flows cannot account for the experimental results. A model consisting of four envelope equations for the fluctuation level, the mean flow shear, the zonal flow amplitude shear, and the averaged pressure gradient is proposed. It is shown that the presence of zonal flows is essential to reproduce the main features of the experimental observations.Comment: 19 pages, 7 figure

Analysis of DIII-D Experiments (DOE GRANT ER54538)

DOE Grant ER5453

Turbulence regulation and stabilization by equilibrium and Time-varying sheared turbulence flows

Turbulence flows are directly measured in a tokamak plasma by applying time-delay-estimation (TDE) analysis to localized 2-D density fluctuation measurements obtained with Beam Emission Spectroscopy on DIII-D. The equilibrium radial flow shear near the plasma edge (0.8 < r/a < 1) varies strongly with magnetic geometry. With the ion grad-B drift directed towards the X-point in a single null plasma, a large radial shear in the poloidal flow is measured, while little shear is observed in the reverse condition. This large shear appears to facilitate the L-to H-mode transition, consistent with the significantly lower LH transition power threshold in this configuration. In addition, time varying, radially localized (k . ρI < 1) flows with a semi-coherent structure peaked near 15 KHz and a very long poloidal wavelength, possibly m=0, are observed. These characteristics are very similar to theoretically predicted zonal flows that are self-generated by and in turn regulate the turbulence

Comparison of L-H transition measurements with physics models

A technique of fitting a modified hyperbolic tangent to the edge profiles has improved the localization of plasma edge parameters. Non-dimensional edge parameters are broadly consistent with several theories of the L-H transition that use edge gradients in their formulation of a critical threshold parameter. The ion {del}B drift direction has only a small effect on the edge plasma conditions measured near the plasma midplane but a large effect on the divertor plasma. The dramatic change of power threshold with the direction of the ion {del}B drift implies that phenomena in the divertor region may be critical for the L-H transition

Local and global Fokker-Planck neoclassical calculations showing flow and bootstrap current modification in a pedestal

In transport barriers, particularly H-mode edge pedestals, radial scale lengths can become comparable to the ion orbit width, causing neoclassical physics to become radially nonlocal. In this work, the resulting changes to neoclassical flow and current are examined both analytically and numerically. Steep density gradients are considered, with scale lengths comparable to the poloidal ion gyroradius, together with strong radial electric fields sufficient to electrostatically confine the ions. Attention is restricted to relatively weak ion temperature gradients (but permitting arbitrary electron temperature gradients), since in this limit a delta-f (small departures from a Maxwellian distribution) rather than full-f approach is justified. This assumption is in fact consistent with measured inter-ELM H-Mode edge pedestal density and ion temperature profiles in many present experiments, and is expected to be increasingly valid in future lower collisionality experiments. In the numerical analysis, the distribution function and Rosenbluth potentials are solved for simultaneously, allowing use of the exact field term in the linearized Fokker-Planck collision operator. In the pedestal, the parallel and poloidal flows are found to deviate strongly from the best available conventional neoclassical prediction, with large poloidal variation of a different form than in the local theory. These predicted effects may be observable experimentally. In the local limit, the Sauter bootstrap current formulae appear accurate at low collisionality, but they can overestimate the bootstrap current near the plateau regime. In the pedestal ordering, ion contributions to the bootstrap and Pfirsch-Schluter currents are also modified