Orbital Magnetism in the Ballistic Regime: Geometrical Effects

We present a general semiclassical theory of the orbital magnetic response of noninteracting electrons confined in two-dimensional potentials. We calculate the magnetic susceptibility of singly-connected and the persistent currents of multiply-connected geometries. We concentrate on the geometric effects by studying confinement by perfect (disorder free) potentials stressing the importance of the underlying classical dynamics. We demonstrate that in a constrained geometry the standard Landau diamagnetic response is always present, but is dominated by finite-size corrections of a quasi-random sign which may be orders of magnitude larger. These corrections are very sensitive to the nature of the classical dynamics. Systems which are integrable at zero magnetic field exhibit larger magnetic response than those which are chaotic. This difference arises from the large oscillations of the density of states in integrable systems due to the existence of families of periodic orbits. The connection between quantum and classical behavior naturally arises from the use of semiclassical expansions. This key tool becomes particularly simple and insightful at finite temperature, where only short classical trajectories need to be kept in the expansion. In addition to the general theory for integrable systems, we analyze in detail a few typical examples of experimental relevance: circles, rings and square billiards. In the latter, extensive numerical calculations are used as a check for the success of the semiclassical analysis. We study the weak-field regime where classical trajectories remain essentially unaffected, the intermediate field regime where we identify new oscillations characteristic for ballistic mesoscopic structures, and the high-field regime where the typical de Haas-van Alphen oscillations exhibit finite-size corrections. We address the comparison with experimental data obtained in high-mobility semiconductor microstructures discussing the differences between individual and ensemble measurements, and the applicability of the present model.Comment: 88 pages, 15 Postscript figures, 3 further figures upon request, to appear in Physics Reports 199

Continuation and stability of rotating waves in the magnetized spherical Couette system: Secondary transitions and multistability

Rotating waves (RW) bifurcating from the axisymmetric basic magnetized spherical Couette (MSC) flow are computed by means of Newton-Krylov continuation techniques for periodic orbits. In addition, their stability is analysed in the framework of Floquet theory. The inner sphere rotates whilst the outer is kept at rest and the fluid is subjected to an axial magnetic field. For a moderate Reynolds number ${\rm Re}=10^3$ (measuring inner rotation) the effect of increasing the magnetic field strength (measured by the Hartmann number ${\rm Ha}$) is addressed in the range ${\rm Ha}\in(0,80)$ corresponding to the working conditions of the HEDGEHOG experiment at Helmholtz-Zentrum Dresden-Rossendorf. The study reveals several regions of multistability of waves with azimuthal wave number $m=2,3,4$, and several transitions to quasiperiodic flows, i.e modulated rotating waves (MRW). These nonlinear flows can be classified as the three different instabilities of the radial jet, the return flow and the shear-layer, as found in previous studies. These two flows are continuously linked, and part of the same branch, as the magnetic forcing is increased. Midway between the two instabilities, at a certain critical ${\rm Ha}$, the nonaxisymmetric component of the flow is maximum.Comment: Published in the Proceedings of the Royal Society A journal. Contains 3 tables and 12 figure

Finite Element Modal Formulation for Panel Flutter at Hypersonic Speeds and Elevated Temperatures

A finite element time domain modal formulation for analyzing flutter behavior of aircraft surface panels in hypersonic airflow has been developed and presented for the first time. Von Karman large deflection plate theory is used for description of the structural nonlinearity and third order piston theory is employed to account for the aerodynamic nonlinearity. The thermal loadings of uniformly distributed temperature and temperature gradients across the panel thickness are incorporated into the finite element formulation. By applying the modal reduction technique, the number of governing equations of motion is reduced dramatically so that the computational time of direct numerical integration is dropped significantly. All possible types of panel behavior, including flat, buckled but dynamically stable, limit cycle oscillation (LCO), periodic motion, and chaotic motion can be observed and analyzed. As examples of the applications of the proposed methodology, flutter responses of isotropic, specially orthotropic and laminated composite panels are investigated. Special emphasis is put on the boundary between LCO and chaos, as well as the routes to chaos. A systematic mode filtering procedure that helps mode selection without specific knowledge of the complex mode shapes is presented and illustrated. Influences of aerodynamic parameters, including aerodynamic damping and Mach number, on the panel flutter responses are studied. The importance of nonlinear aerodynamic terms is examined in detail. The supporting conditions and panel aspect ratio on the onset condition of chaos are also investigated as an illustration of optimization among different design options. Several mathematical tools, including the time history, phase plane plot, Poincaré map, and bifurcation diagram are employed in the chaos study. The largest Lyapunov exponent is also evaluated to assist in detection of chaos. It is found that at low or moderately high nondimensional dynamic pressures, the fluttering panel typically takes a period-doubling route to evolve into chaos, whereas at high nondimensional dynamic pressure, the route to chaos generally involves bursts of chaos and rejuvenations of periodic motions. Various bifurcation behaviors, such as the Hopf bifurcation, pitchfork bifurcation, and transcritical bifurcation, are observed. On the basis of the successful applications presented, the proposed finite element time domain modal formulation and the mode filtering procedure have proven to be an efficient and practical design tool for designers of hypersonic vehicle

Classical and quantum mechanics with chaos

This thesis is concerned with the study, classically and quantum mechanically, of the square billiard with particular attention to chaos in both cases. Classically, we show that the rotating square billiard has two regular limits with a mixture of order and chaos between, depending on an energy parameter, E. This parameter ranges from -2w(^2) to oo, where w is the angular rotation, corresponding to the two integrable limits. The rotating square billiard has simple enough geometry to permit us to elucidate that the mechanism for chaos with rotation or curved trajectories is not flyaway, as previously suggested, but rather the accumulation of angular dispersion from a rotating line. Furthermore, we find periodic cycles which have asymmetric trajectories, below the value of E at which phase space becomes disjointed. These trajectories exhibit both left and right hand curvatures due to the fine balance between Centrifugal and Coriolis forces. Quantum mechanically, we compare the spectral analysis results for the square billiard with three different theoretical distribution functions. A new feature in the study is the correspondence we find, by utilising the Berry-Robnik parameter q, between classical E and a quantum rotation parameter w. The parameter q gives the ratio of chaotic quantum phase volume which we can link to the ratio of chaotic phase volume found classically for varying values of E. We find good correspondence, in particular, the different values of q as w is varied reflect the births and subsequent destructions of the different periodic cycles. We also study wave packet dynamics, necessitating the adaptation of a one dimensional unitary integration method to the two dimensional square billiard. In concluding we suggest how this work may be used, with the aid of the chaotic phase volumes calculated, in future directions for research work

Study of ergodic divertor edge density regimes on the tokamaks Tore Supra and TEXTOR, and sensitivity of tunnel probe electron temperature measurements to a suprathermal electron component

Controlled thermonuclear fusion offers one possible option to meet our future energy needs in a sustainable way. Magnetic confinement in a so-called 'tokamak'-machine is a possible approach towards the achievement of a burning plasma. An important issue in this tokamak research is the transition of the plasma edge to the inner wall. A first topic that is addressed in this thesis, is the ergodic divertor (ED) configuration. An ED achieves the transition between the confined plasma and the wall in a layer where the flux lines have been ergodized by a proper resonant magnetic perturbation. The connection between up- and downstream plasma parameters during ED operation in the tokamaks Tore Supra en TEXTOR has been investigated experimentally by means of Langmuir probes. As an important first step in the theoretical interpretation of those experiments, a Hamiltonian field line mapping code, which had been previously developed for the TEXTOR dynamic ergodic divertor, has been adapted to the geometry of the Tore Supra ED. Subsequently, this adapted code has been used to study some of the properties of the Tore Supra ED magnetic field line structure, as well as to make a qualitative comparison of the sensitivity of the TEXTOR and Tore Supra ergodic divertor magnetic topology to changes in the central density. A second topic of this thesis concerns certain interpretation issues regarding the current-voltage characteristics obtained by a newly developed type of Langmuir probe for the investigation of edge plasmas. More in particular, the sensitivity of the TP to a small population of nonthermal electrons has been investigated in addition to the influence of suprathermal electrons on the scaling and structure of the Debye and the magnetic sheath at the inside of the tunnel probe

Rydberg atoms in strong fields : a testing ground for quantum chaos

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 1995.Includes bibliographical references (p. 279-290).by Michael Coutney.Ph.D

Reconciliation of almost-invariant tori in chaotic systems

Magnetic field lines within toroidal magnetic confinement systems can be described as orbits of one-and-a-half-degree-of-freedom Lagrangian and Hamiltonian systems. In axisymmetric devices such as ideal tokamaks in equilibrium, all field lines lie within a smoothly nested set of invariant tori (magnetic surfaces) foliating the plasma volume, but this integrability is lost within non-axisymmetric devices such as stellarators. That is, not all field lines lie within magnetic surfaces and thus they cannot be described by conventional action-angle coordinates. However, according to the Kolmogorov-Arnold-Moser (KAM) theorem, some invariant tori, covered ergodically by quasiperiodic orbits, survive perturbation away from integrability. Furthermore, by the Poincare-Birkhoff theorem, two kinds of periodic orbits (closed field lines), called the action-minimax and action-minimising orbits, survive perturbation, and they can be incorporated into families of pseudo-orbits that provide best approximations to invariant tori within nonintegrable systems. The three candidates for such almost-invariant tori discussed in this thesis are quadratic-flux-minimising (QFMin) tori, which minimise the integral of the square of the action gradient over the poloidal and toroidal angles, action-gradient-minimising (AGMin) tori, which minimise the square of the action gradient over each periodic pseudo-orbit, and ghost tori, which join the action-minimax and action-minimising orbits via an action-gradient flow. Although none of these almost-invariant tori has any direct physical interpretation, it has been shown by Hudson and Breslau (Phys. Rev. Lett., 100, 095001 (2008)) that ghost tori are in very close correspondence with temperature isotherms. There is also a very close relationship between QFMin, AGMin and ghost tori, which suggests that they could be made equivalent to each other or ""reconciled"" by making an appropriate coordinate transformation. By using the standard map as a discrete-time model for magnetic field lines via the kicked rotor, it is demonstrated using Mathematica that QFMin and ghost tori can be reconciled with each other up to at least k = 1.0 for all rational rotational transforms with denominator less than or equal to 13, where k denotes the nonlinearity parameter of the standard map. This is accomplished by expanding the coordinate transformation as a Fourier series and formulating a variational principle, which is used to numerically construct a set of almost-invariant curves that have rotational transforms equal to continued-fraction convergents of two minus the golden mean (i.e., 0.381966...). By calculating the flux between the curves, it is shown via a preliminary investigation that the reconciled QFMin-ghost tori are consistent with Greene's residue criterion and the existence of KAM tori within the standard map for k < 0.971635..., which is the value at which the last KAM curves are destroyed. It is also shown via the construction of a rigidity principle that the reconciled QFMin-ghost tori have also been reconciled with AGMin tori, by appealing to the fact that discrete-time dynamical systems are merely just Poincare sections of continuous-time dynamical systems, for which the rigidity principle was formulated. This provides an important first step towards constructing an almost-straight-field-line coordinate system for magnetic islands

Classical and quantum chaos of dynamical systems: rotating billiards

The theory of classical chaos is reviewed. From the definition of integrable systems using the Hamilton-Jacobi equation, the theory of perturbed systems is developed and the Kolmogorov-Arnold-Moser (KAM) theorem is explained. It is shown how chaotic motion in Hamiltonian systems is governed by the in tricate connections of stable and unstable invariant manifolds, and how it can be catagorised by algorithmic complexity and symbolic dynamics, giving chaotic measures such as Lyapunov exponents and Kolmogorov entropy. Also reviewed is Gutzwiller's semiclassical trace formula for strongly chaotic systems, torus quantisation for integrable systems, the asymptotic level density for stationary billiards, and random matrix theories for describing spectral fluctuation properties. The classical theory is applied to rotating billiards, particularly the free motion of a particle in a circular billiard rotating uniformly in its own plane about a point on its edge. Numerically, it is shown that the system's classical behaviour ranges from fully chaotic at intermediate energies, to completely integrable at very low and very high energies. It is shown that the strong chaos is due to discontinuities in the Poincare map, caused by trajectories which just glance the boundary-an effect of the curvature of trajectories. Weaker chaos exists due to the usual folding and stretching of the Hamiltonian flow. Approximate invariant curves for integrable motion are found, valid far from the presence of glancing trajectories. The major structures of phase space are investigated: a fixed point and its bifurcation into a two-cycle, and their stabilities. Lyapunov exponents for trajectories are calculated and the chaotic volume for a wide range of energies is measured. Quantum mechanically, the energy spectrum of the system is found numerically. It is shown that at the energies where the classical system is completely integrable the levels do not repel, and at those energies where it is completely chaotic there is strong level repulsion. The nearest neighbour level spacing distributions for various ranges of energy and values of Planck's constant are found. In the semiclassical limit, it is shown that, for energies where the classical system is completely chaotic, the level spacing statistics are Wigner, and where it is completely integrable, the level spacing statistics are Poisson. A model is described for the spacing distributions where the levels can be either Wigner or Poisson, which is useful for showing the transition from one to the other, and ad equately describes the statistics. Theoretically, the asymptotic level density for rotating billiards is calculated, and this is compared with the numerical results with good agreement, after modification of the method to include all levels

Three-dimensional modeling of plasma transport in the HIDRA stellarator

In the edge region of modern nuclear fusion experiments, the interactions between edge plasmas and the materials which ultimately confine them have become increasingly more important as device sizes and powers trend upwards. Devices such as MPEX and PISCES are built to investigate these interactions at the high energies and particle fluxes ejected by transient edge disruption events, but are ultimately linear devices. To extend the diagnostic environments of the greater fusion community to include a dedicated toroidal plasma-material interface (PMI) device, the Hybrid Illinois Device for Research and Applications (HIDRA) has been dedicated. HIDRA is an l=2, m=5 classical stellarator originally built in in the 1970's for RF heating studies. Its most recent users at the Max Planck Institute for Plasma Physics ran the device as WEGA to test heating schemes and train personnel for the recently-completed W-7X advanced stellarator, after which it was gifted to the University of Illinois at Urbana-Champaign. To improve the theoretical models of the PMI environment in the edge region of HIDRA, computational tools which apply these models are required. Many simulation tools currently in use focus on tokamak magnetic geometries or high-power, fully-ionized devices, necessitating the creation of an integrated suite of codes to handle partial ionization with more disparate operational power conditions and classical diffusivity unique to HIDRA in contemporary devices. To this end, HIDRAmod has been created, encompassing the existing coupled codes Edge Monte Carlo 3D (EMC3) and EIRENE to solve the plasma and neutral transport equations. FIELDLINES has been used in the creation of a field-aligned tetrahedral mesh generation utility TORMESH also integrated into HIDRAmod. In addition to these established codes, utilities for calculating the limiting surface and tetrahedral mesh intersection and for post-processing have been written. EMC3 has been altered to include a Bohm-like diffusivity to handle the uniquely diffusive plasma in a self-consistent manner. Preceding operational data on the device, simulations have been run under the context of bounding the incident heat and particle fluxes onto the limiting surface into a region of confidence based on parameters from previous operational campaigns. An outboard midplane limiter, inboard midplane limiter, and 'trench' limiter (along the bottom of the torus) were tested with RF input to core-edge power deposition efficiencies of 10-50% for a 26 kW 2.45 GHz combined RF input discharge. Axial magnetic field strengths of 87.5 mT and 0.5 T were analyzed, corresponding to two heating schemes tested at WEGA. Electron temperatures and densities were seen to match previous WEGA results of 8-10 eV and 1-3 · 10^12 cm^-3 in the edge region respectively. With these results, 26 kW of operational power translates to heat fluxes of up to 1 MW m^-2 on the inboard limiter, up to 0.2 MW m^-2 on the outboard limiter, and up to 0.15 MW m^-2 on the trench limiter. Particle fluxes have been similarly bound by upper limits of 4.7 · 10^22 m^-2 · s^-1, 5 · 10^21 m^-2 · s^-1, and 5.6 · 10^21 m^-2 · s^-1 for the inboard, outboard and trench limiters respectively. Scaling laws for peak electron temperature, Bohm-like diffusivity, and heat and particle fluxes have been calculated for both low- and high-field discharges; peak electron temperatures, particle diffusivity, and heat fluxes at the outboard limiter were seen to follow approximately a power-law of type f(P_RF) ∝ a · P_RF^b, with typical exponents in the range b ∼ 0.55 - 0.60. Higher magnetic fields have the tendency to linearize the heat flux dependence upon the RF power, with exponents in the range of b ∼ 0.75. Particle fluxes on the outboard limiter are seen to saturate first, and then slightly decline for RF powers in excess of 120 kW in the low-field case and 180 kW in the high-field case. Finally, extensions and applications of HIDRAmod are examined, including a path to a self-consistent full-device model and potential optimization strategies which may be employed to enhance fluxes arriving at the limiting surfaces