Ballistic Orbits and Front Speed Enhancement for ABC Flows

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the non-integrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular, the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.Comment: 39 pages, 26 figure

Lagrangian, Game Theoretic and PDE Methods for Averaging G-equations in Turbulent Combustion: Existence and Beyond

G-equations are popular level set Hamilton-Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly non-convex and nonlinear. In the absence of coercivity and convexity, PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging non-coercive, non-convex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.Comment: 69 page

Chemical front propagation in periodic flows: FKPP vs G

We investigate the influence of steady periodic flows on the propagation of chemical fronts in an infinite channel domain. We focus on the sharp front arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the limit of small molecular diffusivity and fast reaction (large P\'eclet and Damk\"ohler numbers, $\mathrm{Pe}$ and $\mathrm{Da}$) and on its heuristic approximation by the G equation. We introduce a variational formulation that expresses the two front speeds in terms of periodic trajectories minimizing the time of travel across the period of the flow, under a constraint that differs between the FKPP and G equations. This formulation shows that the FKPP front speed is greater than or equal to the G equation front speed. We study the two front speeds for a class of cellular vortex flows used in experiments. Using a numerical implementation of the variational formulation, we show that the differences between the two front speeds are modest for a broad range of parameters. However, large differences appear when a strong mean flow opposes front propagation; in particular, we identify a range of parameters for which FKPP fronts can propagate against the flow while G fronts cannot. We verify our computations against closed-form expressions derived for $\mathrm{Da}\ll \mathrm{Pe}$ and for $\mathrm{Da}\gg \mathrm{Pe}$

Monte Carlo simulation of the Jovian plasma torus interaction with Io’s atmosphere and the resultant aurora during eclipse

textIo, the innermost Galilean satellite of Jupiter, exhibits a wide variety of complex phenomena such as interaction with Jupiter’s magnetosphere, volcanic activity, and a rarefied multi-species sublimating and condensing atmosphere with an ionosphere. Io’s orbital resonance with Jupiter and the other Galilean satellites produces intense tidal heating. This makes Io the most volcanically active body in the solar system with plumes that rise hundreds of kilometers above the surface. In the present work, the interaction of Io’s atmosphere with the Jovian plasma torus is simulated via the Direct Simulation Monte Carlo (DSMC) method and the aurora produced via electron-neutral excitation collisions is examined using electron transport Monte Carlo simulation. The electron-transport Monte Carlo simulation models the electron collisions with the neutral atmosphere and their transport along field lines as they sweep past Io, using a pre-computed steady atmosphere and magnetic field. As input to the Monte Carlo simulation, the neutral atmosphere was first modeled using prior 2D sunlit continuum simulations of Io’s atmosphere produced by others. In order to justify the use of a sunlit atmosphere for eclipse, 1D two-species (SO2 and a non-condensable) DSMC simulations of Io’s atmospheric dynamics during and immediately after eclipse were performed. It was found that the inclusion of a non-condensable species (SO or O2) leads to the formation of a diffusion layer which prevents rapid collapse. The degree to which the diffusion layer slowed the atmospheric collapse was found to be extremely sensitive to both the initial non-condensable mole fraction and the reaction (or sticking) probability on the surface of the “non-condensable”. Furthermore, upon egress, vertical stratification of the atmosphere occurred with the non-condensable species being lifted to higher altitudes by the rapid sublimation of SO2 as the surface warms. Simulated aurorae (specifically the [OI] 6300 Å and the S2, SO, and SO2 molecular band emission in the middle ultraviolet) show good agreement with observations of Io in eclipse and an attempt was made to use the simulations to constrain the upstream torus electron temperature and Io’s atmospheric composition, structure, and volcanic activity. It is found that the position of the bright [OI] 6300 Å wake spot relative to Io’s equator depends on the position of Io relative to the plasma torus’ equator and the asymmetric electron number flux that results. Using HST/STIS UV-Vis spectra, the upstream electron temperature is weakly constrained to be between 3 eV and 8 eV depending on the flux of a low energy (35 eV), non-thermal component of the plasma (more non-thermal flux requires lower thermal plasma temperatures to fit the spectrum). Furthermore, an upper limit of 5% of the thermal torus density (or 180 cm−3 based on the Galileo J0 plasma density at Io) is obtained for the low energy non-thermal component of the plasma. These limits are consistent with Galileo observations of the upstream torus temperature and estimates for the the non-thermal component. Finally, plume activity and S2 content during eclipse observations with HST/STIS were constrained by examining the emission intensity along the spatial axis of the aperture. During the August 1999 UV-Vis observations, the auroral simulations indicate that the large volcanoes Pele and Surt were inactive whereas Tvashtar was active and that Dazhbog and possibly Loki were also actively venting gas. The S2 content inferred for the large Pele-type plumes was between 5% (Tvashtar) and 30% (Loki, if active), consistent with prior observations (Spencer et al., 2000; Jessup et al., 2007). A 3D DSMC simulation of Io’s sublimation and sputtered atmosphere including photo- and plasma-chemistry was developed. In future work these atmospheric simulations will replace the continuum target atmosphere in the auroral model and thus enable a better match to the observed high altitude auroral emission. In the present work, the plasma interaction is modeled by a flux of ions and electrons which flow around and through Io’s atmosphere along pre-computed fields and interact with the neutral gas. A 3D DSMC simulation of Io’s atmosphere assuming a simple thermal model for the surface just prior to ingress into eclipse and uniform frost coverage has been performed in order to understand how Io’s general atmospheric dynamics are affected by the new plasma model with chemistry and sputtering. Sputtering was found to supply most of the nightside atmosphere (producing an SO2 column of ~5×1013 cm−2); however, the dense dayside sublimation atmosphere was found to block sputtering of the surface. The influence of the dynamic plasma pressure on the day-to-night circumplanetary flow was found to be quite substantial causing the day-to-night wind across the dawn terminator to flow slightly towards the equator. This results in a region of high density near the equator that extends far (~2000 km for the condensable species) onto the nightside across the dawn terminator. Thus, even without thermal lag due to rotation or variable surface frost, highly asymmetric equatorial column densities relative to the subsolar point are obtained. The non-condensable O2, which is a trace species on the dayside, is the dominant species on the nightside despite increased SO2 sputtering because the loss rate of O2 is slow. Finally, a very intriguing O2 flow feature was observed near the dusk terminator where the flow from the leading hemisphere (pushed by the plasma) meets the flow from the dayside trailing hemisphere. Since the O2 does not condense on the surface, it slowly convects towards the poles and then back onto the nightside, eventually to be dissociated or stripped away by the plasma.Aerospace Engineerin

Quantum properties of atomic-sized conductors

Using remarkably simple experimental techniques it is possible to gently break a metallic contact and thus form conducting nanowires. During the last stages of the pulling a neck-shaped wire connects the two electrodes, the diameter of which is reduced to single atom upon further stretching. For some metals it is even possible to form a chain of individual atoms in this fashion. Although the atomic structure of contacts can be quite complicated, as soon as the weakest point is reduced to just a single atom the complexity is removed. The properties of the contact are then dominantly determined by the nature of this atom. This has allowed for quantitative comparison of theory and experiment for many properties, and atomic contacts have proven to form a rich test-bed for concepts from mesoscopic physics. Properties investigated include multiple Andreev reflection, shot noise, conductance quantization, conductance fluctuations, and dynamical Coulomb blockade. In addition, pronounced quantum effects show up in the mechanical properties of the contacts, as seen in the force and cohesion energy of the nanowires. We review this reseach, which has been performed mainly during the past decade, and we discuss the results in the context of related developments.Comment: Review, 120 pages, 98 figures. In view of the file size figures have been compressed. A higher-resolution version can be found at: http://lions1.leidenuniv.nl/wwwhome/ruitenbe/review/QPASC-hr-ps-v2.zip (5.6MB zip PostScript

Quantum anomalous hall effect, domain walls, and disorder in bilayer graphene

Seit seiner Entdeckung im Jahr 2004 ist das zweidimensionale Material Graphen Gegenstand vieler theoretischer sowie experimenteller Studien, wobei außergewöhnliche mechanische und elektrische Eigenschaften entdeckt wurden. Im Vergleich zur Monolage zeichnet sich Bilagen Graphen durch ähnlich herausragende Qualitäten aus, besitzt dabei aber noch größere Vielseitigkeit, beispielsweise durch eine variierbare Bandlücke. Zudem ist Bilagen Graphen, auf Grund seiner unter gewissen Umständen nicht verschwindenden Zustandsdichte bei Ladungsneutralität, besonders anfällig für korrelierte Zustände. Diese treten durch Elektron-Elektron Wechselwirkungen auf, wobei bestimmte Symmetrien des Systems gebrochen werden und sich das Energiespektrum verändert. Theoretische Studien nennen beispielsweise fünf verwandte Quanten-Hall-Zustände, die durch Brechung der chiralen Symmetrie entstehen können und bei Ladungsneutralität miteinander konkurrieren. Obwohl nach und nach einige dieser Zustände durch die immer besser werdende Qualität der Proben experimentell bestätigt werden konnten, gibt es diesbezüglich noch viele offene Fragestellungen. Insbesondere konnte einer dieser Quanten-Hall-Zustände, die exotische „ALL“-Phase, welche eine teilweise Polarisierung der zum Transport beitragenden Ladungsträger in eine der Graphenlagen und ein orbitales magnetischen Moment aufweist, bisher noch nicht eindeutig beobachtet werden. Des Weiteren ist bisher noch weitestgehend unklar, welche der fünf Quanten-Hall-Phasen der eigentliche Grundzustand von Bilagen Graphen ist, da die bis zum jetzigen Zeitpunkt veröffentlichten Studien keine eindeutigen experimentellen Beobachtungen liefern. Neben dem Auftreten von konkurrierenden Quanten-Hall-Zuständen könnte die Existenz von Fehlern in der Stapelfolge der zwei Graphenlagen eine mögliche Erklärung für die unterschiedlichen Signaturen in Quantentransportmessungen sein. Die Detektion dieser Kristallfehler wurde erst vor Kurzem durch präzise Techniken, wie beispielsweise optische Rasternahfeldmikroskopie, ermöglicht. Obwohl schon eindrucksvoll quantisierter Ladungstransport entlang solcher Kristallfehler im Experiment gezeigt wurde, bleibt ihr Einfluss auf die bei Ladungsneutralität auftretenden Quanten-Hall-Zustände weitestgehend unerforscht. Um die aufgeführten Fragestellungen genauer zu untersuchen, werden in dieser Arbeit Quantentransportmessungen in Bilagen Graphen bei niedrigen Temperaturen präsentiert. Diese wurden an Feldeffekttransistoren, bestehend aus ultrareinem, freischwebenden Bilagen Graphen, dessen elektrische Eigenschaften durch zwei Gate-Elektroden manipulierbar sind, durchgeführt. Besonderes Augenmerk wurde dabei auf die Existenz von Fehlern in der Stapelfolge innerhalb der untersuchten Graphen Flocken gelegt. Sind diese nicht vorhanden, konnte die exotische „ALL“-Phase bei niedrigen Magnetfeldern beobachtet werden, wobei der Zustand in achtfacher Ausführung in Form eines anomalen Quanten-Hall-Effekts mit einer Leitfähigkeit von ±2 e^2 h^(-1) (e ist dabei die Elementarladung und h das Plancksche Wirkungsquantum) auftritt. Die Entdeckung stellt einen überzeugenden Nachweis für orbitalen Magnetismus in Bilagen Graphen dar und verdeutlicht, dass das vermeintlich triviale System einen anomalen Quanten-Hall-Effekt aufweist, ohne dass die Realisierung eines fragilen Moiré-Gitters notwendig ist. Außerdem wurde der Quantentransport entlang Fehlern in der Stapelfolge von Bilagen Graphen untersucht. Dabei wurde ein komplexes Zusammenspiel zwischen topologisch geschütztem Quantentransport entlang eines Kristallfehlers und Quantentransport in Randkanälen, induziert durch den Quanten-Hall-Effekt, entdeckt. Die Messungen zeigen den maßgeblichen Einfluss der häufig vorkommenden Kristallfehler und verdeutlichen, wie wichtig es ist, diesen in zukünftigen Studien zu beachten. Zuletzt wurden die Auswirkungen von Unordnung sowie Fehlern in der Stapelfolge auf den Grundzustand und auf verschiedene Phasenübergänge zwischen Zuständen mit gebrochener Symmetrie in Bilagen Graphen untersucht. Die Ergebnisse helfen schwer erklärbare Signaturen in Quantentransportmessungen aus der Literatur zu verstehen und tragen somit zur eindeutigen Identifikation des Grundzustands von Bilagen Graphen bei. Durch die hier präsentierten Ergebnisse wurden bedeutende Fortschritte im Verständnis komplexer physikalischer Phänomene in Bilagen Graphen erzielt, was zudem die Wichtigkeit weiterer experimenteller Studien an dem Material verdeutlicht.Since the discovery of graphene in 2004, the two-dimensional material has been subject of extensive theoretical and experimental research revealing exceptional electronic and mechanical properties. Bilayer graphene, while inheriting most advantages of its monolayer counterpart, provides even more tunability, e.g. due to its tunable band gap. Moreover, as consequence of the non-vanishing density of states near charge neutrality under certain circumstances, bilayer graphene is susceptible to exotic interaction-driven broken-symmetry states that modify the energetic spectrum. For example, theoretical studies propose the emergence of a family of five competing quantum Hall states at charge neutrality owing to chiral symmetry breaking. Although some of the phases have already been observed experimentally with an increasing level of device quality, bilayer graphene retains many related unanswered questions. For instance, the exotic ALL phase, a quantum anomalous Hall phase with partial layer polarization and substantial orbital moment, has not been pinpointed clearly. Moreover, it is still under debate which of the five broken-symmetry phases is the true ground state, as ambiguous experimental results have been reported from literature. Besides the emergence of competing phases, a possible cause for distinct signatures in quantum transport measurements could be the influence of stacking domain walls in bilayer graphene. Their detection has only become possible recently using precise scanning techniques such as scattering-type scanning near-field optical microscopy. Although quantum transport along such dislocations has been shown, their impact on broken-symmetry states emerging within the zero energy Landau level remains unclear. To shed light on these unexplored aspects, low-temperature transport measurements on high-quality dually gated freestanding bilayer graphene are presented in this thesis, with special attention given to any stacking domain walls present within the bilayer graphene flakes. In their absence, the exotic ALL phase, appearing as an octet of quantum anomalous Hall phases with a conductance of ±2 e^2 h^(-1) (where e is the electronic charge and h is Planck’s constant), was tracked to low magnetic fields, providing compelling evidence for orbital magnetism in bilayer graphene. The findings demonstrate that the seemingly simple Bernal-stacked bilayer graphene exhibits the quantum anomalous Hall effect without the need of fabricating delicate moiré heterostructures. In addition, the quantum transport along stacking domain walls was investigated revealing an intriguing interplay between topological valley and quantum Hall edge transport. The measurements highlight the influence of the commonly occurring stacking domain walls and demonstrate that their impact inevitably needs to be regarded in future experiments. Lastly, the role of disorder and stacking domain walls on the emergence of the spontaneously gapped ground state and various phase transitions between broken-symmetry states was examined. The results contribute to solving the debate about the ground state of bilayer graphene and help to explain related ambiguous observations in literature. All in all, the presented measurements provide major advances in understanding the complex physical phenomena in the seemingly trivial Bernal-stacked bilayer graphene and highlight the importance of continuous experimental effort