Large-amplitude State Research Articles

Existence of breathing patterns in globally coupled finite-size nonlinear lattices

ABSTRACTWe prove the existence of time-periodic solutions representing breathing patterns in general nonlinear Hamiltonian finite-size lattices with global coupling. As a first step the existence of localised solutions of a two site segment, where one oscillator performs larger-amplitude motion compared to the other one, is established. To this end the existence problem is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem. For isoenergetic states the degree of localisation can be tuned in a wide range by changing the initial data and the coupling strength. Subsequently, it is proven that a related localised state can be excited in the extended nonlinear lattice forming a breathing pattern with a single site of large amplitude against a background of uniform small-amplitude states. Furthermore, it is demonstrated that also spatial patterns are possible that are built up from any combination of the small-amplitude state and the large-amplitude state.

Designing graphene structures with controlled distributions of topological defects: A case study of toughness enhancement in graphene ruga

A novel design methodology combining phase field crystal method and atomistic simulations is proposed to solve the inverse problem of finding the optimized distribution and type of topological defects that make a graphene sheet conform to a targeted arbitrary three dimensional (3D) surface. To demonstrate potential applications of the proposed method, we created a sinusoidal graphene structure with wavelength of 4 nm and amplitude of 0.75 nm, and then demonstrated using large-scale molecular dynamics (MD) simulations that the constructed graphene ruga11The Latin word ruga is used to refer to a large-amplitude state of wrinkle, crease, ridge or fold [1]. has a fracture toughness around 25J/m2, which is about twice that of the defect-free graphene. The underlying toughening mechanisms include nanocrack shielding and atomic scale crack bridging. This study suggests a promising general methodology to tailor-design mechanical properties of graphene through controlled distributions of topological defects.

Stability Analysis of a Droplet Pinned in Channel Under Gravity

The stability of a two-dimensional, incompressible droplet, with two cylindrical-caps that is held in a channel under gravity, is investigated through the development of an analytical model based on the Young–Laplace relationship. The droplet state is measured by the location of its center of mass, where the center of mass is derived analytically by assuming a circular shape for the droplet cap. The derived analytical expressions are validated through the use of computational fluid dynamics (CFD). When a droplet is suspended under no gravity conditions, there is a critical droplet volume Vcr where asymmetric droplet states appear in addition to the basic symmetric states when the drop volume V > Vcr. When V < Vcr, the symmetric droplet states are stable, and when V > Vcr, the symmetric states are unstable and the asymmetric states are stable. With gravity, the pitchfork bifurcation diagram of the droplet system changes into two separate branches of equilibrium states: The primary branch describes a gradual and stable change of the droplet from a symmetric to asymmetric state as the droplet volume is increased. The secondary branch appears at a modified critical volume Vmcr and describes two additional asymmetric states when V > Vmcr. The large-amplitude states along the secondary branch are stable whereas the small-amplitude states are unstable. There exists a maximum volume on each of the primary and secondary branch where the droplet no longer sustains its weight and where the maximum volume on the primary branch is smaller than the maximum volume on the secondary branch. There is a critical value for the strength of the gravity force, relative to the capillary force, that provides the condition at which a droplet state exists only at the primary branch; the secondary branch is unstable. Analytical solutions show good agreement with CFD results as long as the circular shape assumption of the droplet cap is approximately valid.

Defects controlled wrinkling and topological design in graphene

Due to its atomic scale thickness, the deformation energy in a free standing graphene sheet can be easily released through out-of-plane wrinkles which, if controllable, may be used to tune the electrical and mechanical properties of graphene. Here we adopt a generalized von Karman equation for a flexible solid membrane to describe graphene wrinkling induced by a prescribed distribution of topological defects such as disclinations (heptagons or pentagons) and dislocations (heptagon–pentagon dipoles). In this framework, a given distribution of topological defects in a graphene sheet is represented as an eigenstrain field which is determined from a Poisson equation and can be conveniently implemented in finite element (FEM) simulations. Comparison with atomistic simulations indicates that the proposed model, with only three parameters (i.e., bond length, stretching modulus and bending stiffness), is capable of accurately predicting the atomic scale wrinkles near disclination/dislocation cores while also capturing the large scale graphene configurations under specific defect distributions such as those leading to a sinusoidal surface ruga22The Latin word ruga is used to refer a large-amplitude state of a wrinkle, crease, ridge or fold (Diab et al., 2013). or a catenoid funnel.

A double-integration hypothesis to explain ocean ecosystem response to climate forcing

Long-term time series of marine ecological indicators often are characterized by large-amplitude state transitions that can persist for decades. Understanding the significance of these variations depends critically on the underlying hypotheses characterizing expected natural variability. Using a linear autoregressive model in combination with long-term zooplankton observations off the California coast, we show that cumulative integrations of white-noise atmospheric forcing can generate marine population responses that are characterized by strong transitions and prolonged apparent state changes. This model provides a baseline hypothesis for explaining ecosystem variability and for interpreting the significance of abrupt responses and climate change signatures in marine ecosystems.

Convectons in periodic and bounded domains

Numerical continuation is used to compute spatially localized convection in a binary fluid with no-slip laterally insulating boundary conditions and the results are compared with the corresponding ones for periodic boundary conditions (PBC). The change in the boundary conditions produces a dramatic change in the snaking bifurcation diagram that describes the organization of localized states with PBC: the snaking branches turn continuously into a large amplitude state that resembles periodic convection with defects at the sidewalls. Odd parity convectons are more affected by the boundary conditions since the sidewalls suppress the horizontal pumping action that accompanies these states in spatially periodic domains.

Spatiotemporal dynamics near the onset of convection for binary mixtures in cylindrical containers

Pattern selection near the onset of convection in a cylindrical container heated from below is investigated numerically for a water-ethanol mixture, with parameter values and boundary conditions relevant to experiments. The Boussinesq three-dimensional equations for binary fluid convection are simulated for cylinders of aspect ratio Gamma=11 and 10.5 (Gamma identical with R/d, where R is the radius of the cell and d its height). The onset of convection occurs via a subcritical Hopf bifurcation in which the critical mode is strongly influenced by small variations of the aspect ratio of the cell. During the linear regime, an m=1 azimuthal mode consisting of radially traveling waves grows in amplitude in the Gamma=11 cell, while an m=0 azimuthal mode is selected in the Gamma =10.5 cylinder. As convection evolves, simulations for subcritical and supercritical Rayleigh numbers reveal differences in the dynamics. Very close to the critical value, convection is erratic and focuses along one or more diameters of the cell; growths and collapses of the convection amplitude take place, but convection eventually dies away for subcritical values and persists for slightly supercritical values. For larger supercritical values, convection grows progressively in amplitude, and patterns consist of traveling-wave regions of convection initially focused near the cell center, though expanding slowly until a large-amplitude state is reached. Depending on the reduced Rayleigh number, the final state can be a nonsteady state filling the cell or a disordered confined state.

Nonlinear Equilibration of Thermohaline Intrusions

Abstract The nonlinear behavior of thermohaline intrusions on a wide front is investigated using a one-dimensional numerical model. The model is used to follow the evolution of a field of intrusions from infinitesimal amplitude to a large-amplitude state characterized by inversions in temperature and salinity. It is thus possible to extend the analytical studies of Toole and Georgi and others to large amplitude, allowing for the effects of amplitude-dependent diffusivities, and for the appearance of stably stratified, diffusively stratified, and statically unstable regions in the water column as the intrusions grow. The model runs are initialized with infinitesimal disturbances that grow exponentially in time until inversions in temperature and salinity occur. After inversions appear, the intrusions evolve toward an equilibrium state in which friction balances buoyancy forces, for both finger- and diffusive-sense basic-state stratifications. These equilibrium states are characterized by statically unstabl...

The influence of surface topography on rotating convection

Busse's annulus is considered as a model of thermal convection inside the Earth's liquid core. The conventional tilted base and top are modified by azimuthal sinusoidal corrugations so that the effects of surface topography can be investigated. The annulus rotates rapidly about its axis of symmetry with gravity directed radially inwards towards the rotation axis. An unstable radial temperature gradient is maintained and the resulting Boussinesq convection is considered at small Ekman number. Since the corrugations on the boundaries cause the geostrophic contours to be no longer circular, strong geostrophic flows may be driven by buoyancy forces and damped by Ekman suction. When the bumps are sufficiently large, instability of the static state is dominated by steady geostrophic flow with the convection pattern locked to the bumps. As the bump size is decreased, oscillatory geostrophic flow is possible but the preferred mode is modulated on a long azimuthal length scale and propagates as a wave eastwards. This mode only exists in the presence of bumps and is not to be confused with the thermal Rossby waves which are eventually preferred as the bump height tends to zero. Like thermal Rossby waves, the new modes prefer to occupy the longest available radial length scale. In this long-length-scale limit, two finite-amplitude states characterized by uniform geostrophic flows can be determined. The small-amplitude state resembles Or & Busse's (1987) mean flow instability. On losing stability the solution jumps to the more robust large-amplitude state. Eventually, for sufficiently large Rayleigh number and bump height, it becomes unstable to a long-azimuthal-length-scale travelling wave. The ensuing finite-amplitude wave and the mean flow, upon which it rides, are characterized by a geostrophic flow, which is everywhere westward.

Phase distributions and large-amplitude states.

We compare the properties of four quantum phase distributions: the London distribution, the integrated Wigner function, the integrated Q function, and the quadrature-based phase distribution. Their utility in determining the accuracy of phase-shift measurements and their transformation properties under rotations and squeezing transformations are considered. We show that two of the distributions become the same for large-amplitude states that are sufficiently localized in phase space. We call these states quasiclassical phase states. Large-amplitude classical states fall into this class. Some restrictions on how peaked phase distributions of classical states can be are discussed. For quasiclassical phase states, the phase distribution shares many of the properties of classical phase distributions and is measurable. For other states, such as small-amplitude or some nonclassical states, there is no phase distribution with all the desired properties.

Vibrational states and structure of Ar3: The role of three-body forces

Vibrational energies and eigenfunctions of Ar3, including some pertaining to highly excited states, are computed, and insights into their dynamical and structural properties are obtained. The method used employs the vibrational self-consistent-field (SCF) theory in hyperspherical coordinates as a first approximation. Exact results are obtained by configuration interaction, using the SCF states as an efficient basis. A focal point of the study is the effect of three-body potentials on the vibrational spectrum. Axilrod–Teller and other three-body potentials are used to examine this. It is found that the effect of three-body forces on the spectrum is substantial, and larger than effects due to uncertainties in the presently known two-body Ar–Ar potentials. This suggests that experimental spectroscopy of Ar3 may be used to determine reliable three-body forces among Ar atoms. It is also shown that the three-body double-dipole–quadrupole interaction, while less important than the Axilrod–Teller one, has a significant effect on the vibrational spectrum. Finally, a detailed analysis is made of the Ar–Ar distance distributions in the various states, of the structural distributions of Ar3, and of the properties of the wave functions. We find that the wave functions show well-ordered nodal patterns even for the highly excited large-amplitude states. Thus, these states do not correspond qualitatively to ‘‘liquid-like’’ behavior of the cluster.