Search

In this paper we consider the motion of point particles in a particular type of one-degree-of-freedom, slowly changing, temporally periodic Hamiltonian. Through most of the time cycle, the particles conserve their action, but when a separatrix is approached and crossed, the conservation of action breaks down, as shown in previous theoretical studies. These crossings have the effect that the numerical solution shows an apparent contradiction. Specifically we consider two initial constant energy phase space curves $H={E}_{A}$ and $H={E}_{B}$ at time $t=0$, where $H$ is the Hamiltonian and ${E}_{A}$ and ${E}_{B}$ are the two initial energies. The curve $H={E}_{A}$ encircles the curve $H={E}_{B}$. We then sprinkle many initial conditions (particles) on these curves and numerically follow their orbits from $t=0$ forward in time by one cycle period. At the end of the cycle the vast majority of points initially on the curves $H={E}_{A}$ and $H={E}_{B}$ now appear to lie on two new constant energy curves $H={E}_{A}^{\ensuremath{'}}$ and $H={E}_{B}^{\ensuremath{'}}$, where the ${B}^{\ensuremath{'}}$ curve now encircles the ${A}^{\ensuremath{'}}$ curve (as opposed to the initial case where the $A$ curve encircles the $B$ curve). Due to the uniqueness of Hamilton dynamics, curves evolved under the dynamics cannot cross each other. Thus the apparent curves $H={E}_{A}^{\ensuremath{'}}$ and $H={E}_{B}^{\ensuremath{'}}$ must be only approximate representations of the true situation that respects the topological exclusion of curve crossing. In this paper we resolve this apparent paradox and study its consequences. For this purpose we introduce a ``robust'' numerical simulation technique for studying the complex time evolution of a phase space curve in a Hamiltonian system. We also consider how a very tiny amount of friction can have a major consequence, as well as what happens when a very large number of cycles is followed. We also discuss how this phenomenon might extend to chaotic motion in higher dimensional Hamiltonian systems.

MARFEs are experimentally studied using spectroscopic techniques to determine their internal characteristics. We find the local density and temperature in the MARFE to be $(2--3)\ifmmode\times\else\texttimes\fi{}{10}^{21}\mathrm{m}{}^{\ensuremath{-}3}$ and 0.7--1.0 eV, respectively. The volume recombination rate is also large in the MARFE, $(5--10)\ifmmode\times\else\texttimes\fi{}{10}^{22}\mathrm{s}{}^{\ensuremath{-}1}$, similar in magnitude to the plasma source rate in the core. Because this plasma sink is located in confined regions of the plasma, the MARFE acts like a limiter; ions ``recycle'' from the MARFE as neutrals. These characteristics have not been predicted by current models.

An alternative type of Hamiltonian nonlinear resonant island is analyzed. In the usual case where the resonant island is pendulumlike, chains bifurcated out of the central elliptic point undergo infinite cascades of period-doubling bifurcations as they approach the island boundary. In the present case we find that those chains undergo either period doubling or inverse saddle-node bifurcations, depending on the strength of perturbing terms. In the saddle-node case we show that just after a reconnection process, external chains cross the island boundary to collapse against the bifurcated internal chains.

The first sustainment of toroidal plasma current of 50 kA at up to 3 times the injected currents, added in quadrature, using steady inductive helicity injection is described. Separatrix currents-currents not linking the helicity injectors-are sustained up to 40 kA. Decreases in the n=1 toroidal mode of the poloidal magnetic field at higher current amplifications indicate more quiescent, direct toroidal current drive. Results are achieved in HIT-SI (with a spheromak of major radius 0.3 m) during deuterium operations immediately after helium operation. These results represent a breakthrough in the development of this new current drive method for magnetic confinement fusion.

With the help of phase-space pictures we consider the dynamics of a hydrogen atom driven by a high-frequency periodic electromagnetic field. We show that the stabilization phenomenon, widely discussed in the literature, has quantum and classical appearances, which correspond to different Physical situations. Classical stabilization relies on the dynamics of the electron in the laser-induced effective potential: the electron repeatedly crosses the separatrix of the Coulomb potential moving along the trajectories of the effective potential

Plasma loss due to apparatus asymmetries is a ubiquitous phenomenon in magnetic plasma confinement. When the plasma equilibrium has locally trapped particle populations partitioned by a separatrix from one another and from passing particles, the asymmetry transport is enhanced. The trapped and passing particle populations react differently to the asymmetries, leading to the standard $1/\ensuremath{\nu}$ and $\sqrt{\ensuremath{\nu}}$ transport regimes of superbanana orbit theory as particles collisionally scatter from one orbit type to another. However, when the separatrix is itself asymmetric, particles can collisionlessly transit from trapped to passing and back, leading to enhanced transport.

The chaotic dynamics in a separatrix layer are investigated using the ``separatrix map.'' Quantizing this map we show that localization effects inhibit the quantum transport across the separatrix layer. Depending on the size of the nonintegrability parameter, the mechanism of localization is connected with either the remnants of the barely broken separatrix, or with barely broken Kolmogorov-Arnold-Moser tori (``cantori'') inside the separatrix layer, or with dynamical localization due to destructive interference of random transition amplitudes.

Weak axial variations in magnetic or electric confinement fields in pure electron plasmas cause slow electrons to be trapped locally, and collisional diffusion across the trapping separatrix then causes surprisingly large trapped-particle-mediated (TPM) damping and transport effects. Here we characterize TPM damping of m{sub {theta}}{ne}0, m{sub z}={+-}1 Trivelpiece-Gould plasma modes in large-amplitude long-lived Bernstein-Greene-Kruskal states. The TPM damping gives {gamma}{sub BGK}/{omega}{approx}10{sup -4} and seems to dominate in regimes of weak interparticle collisions.

Asymptotical behavior of canonical mappings near the separatrix of Hamiltonian systems subjected to time-periodic perturbations is studied. Based on general forms of these mappings [S. S. Abdullaev, Phys. Rev. E 70, 046202 (2004)] it is shown that the Melnikov-type integrals determining their generating functions can be presented as a sum of regular, ${R}^{(\mathit{reg})}(h)$, and oscillatory, ${R}^{(\mathit{osc})}(h)$, parts. General asymptotical formulas for ${R}^{(\mathit{osc})}(h)$ are derived. The oscillatory parts have zeros at primary resonant values of energy. Conditions are found at which the oscillatory parts, ${R}^{(\mathit{osc})}(h)$, can be neglected in the generating functions thus allowing us to obtain simplified mappings depending only the regular parts ${R}^{(\mathit{reg})}(h)$. Since the latter are smooth functions of energy $h$ this allows us also to justify the widely used conventional separatrix mapping determined by ${R}^{(\mathit{reg})}(h)$ at the separatrix $h=0$. A theory is illustrated for a specific example of a Hamiltonian system, a particle dynamics in periodically perturbed double-well potential.

We show that the onset of global chaos in a time periodically perturbed Hamiltonian system may occur at unusually small magnitudes of perturbation if the unperturbed system possesses more than one separatrix. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with separatrices of the unperturbed system. We develop the asymptotic theory and verify it in simulations.

The transition from the low to the high mode of plasma confinement ( $L\ensuremath{-}H$ transition) is studied in the DIII-D by an experimental technique which allows an arbitrarily slow transition. During an initial transition, periodic turbulent instability bursts are observed near the separatrix which inhibit the full transition. These bursts are damped by self-generated shear flows, and a predator-prey-type relationship is shown to give a good description of the data. As the neutral-beam power is raised, the oscillations change to type III edge localized modes. Another transition then leads to a quiet $H$ mode.

We have investigated the chaotic and frequency-locked population oscillations between two coupled Bose-Einstein condensates with time-dependent asymmetric potential and damping. Under the deterministic perturbation, there exist stable oscillations close to the separatrix solution, which are Melnikov chaotic. Numerical results reveal that, in the nondissipative regime, regular oscillations gradually tend to chaotic with the increase of the trap asymmetry, the long-term localization disappears, and short-term localization can be changed from one of the Bose-Einstein condensates to the other through the route of Rabi oscillation. But in the dissipative regime, stationary chaos disappears and transient chaos is a common phenomenon before the regular stable frequency-locked oscillations, and a proper damping can keep the localization long lived.

Wall stabilization of the free boundary $n=1$ magnetohydrodynamic modes of the spheromak has been studied numerically. Line tying of the open field lines has been found to improve stability dramatically. The dependence of stability on separatrix shape and current profile has also been determined.

To study the stationary probability distribution of master equations in the large-system-size limit, we introduce a kind of master Hamiltonian which is similar to the Fokker-Planck Hamiltonian suggested by Graham et al. in the study of Fokker-Planck equations in the weak-noise limit. The general procedure to associate the potential of the master equation with a certain separatrix of the master Hamiltonian system is described. With some solvable models we show how the smoothness of the potential of the master equation is related to the integrability of the corresponding master Hamiltonian system.

The change of the crossing parameter (essentially the phase) between sequential slow separatrix crossings is calculated for Hamiltonian systems with one degree of freedom. Combined with the previous separatrix crossing analysis, these results reduce the dynamics of adiabatic systems with separatrices to a map. This map determines whether a trajectory leaving a given separatrix lobe is ultimately captured by the other lobe. Averaging these results over initial phase yields the reaction probability, which does not asymptote to the fully phase-mixed result even for arbitrarily long times between separatrix crossings.

High-..beta.., collisionless, toroidal multipole equilibria with parameters n = 5 x 10/sup 13/ cm/sup -3/, T/sub i/ = 200 eV, T/sub e/ = 25 eV, and bridge ..beta.. = 8% have been obtained in the University of California at Los Angeles dodecapole. The equilibrium is characterized by perturbed vacuum magnetic fields, flat internal pressure profile with sharp edge, and a steep hoop bridge profile 2 ion-gyroradii wide centered on the separatrix. The transition from diffuse profiles at low ..beta.. or =5% is observed. A stable equilibrium is observed in excess of 400 Alfven times.

The three-dimensional stability of magnetic equilibria with islands is investigated with use of reduced magnetohydrodynamics. It is found numerically that tokamak equilibria with large $\frac{m}{n}=2$ islands can be ideally unstable. The instability is explained by the strongly modified $q$ profile in the helical equilibrium, where the $q=\frac{3}{2},\frac{5}{3},\dots{}$ resonant surfaces are all close to the inner separatrix of the islands. An analytic treatment of the three-dimensional coalescence instability is given to illustrate this effect.