Finite Spatial Extent Research Articles

Compact-like kink in a real electrical reaction–diffusion chain

Abstract We demonstrate experimentally the compact-like kinks existence in a real electrical reaction–diffusion chain. Our measures show that such entities are strictly localized and consequently present a finite spatial extent. We show equally that the kink velocity is threshold-dependent. A theoretical quantification of the critical coupling under which propagation fails is also achieved and reveals that nonlinear coupling leads to a propagation failure reduction.

Switching Behaviour of Magnetic Particles with DipolarInteraction

We study the switching in the magnetic moments of interactingmagnetic particles. The dynamics of the magnetic moments is governedby a coupled set of Landau–Lifshitz–Gilbert equations. The magneticparticles are assumed to be spherical in shape, single domain, and haveuniaxial anisotropy. Effects of dipolar interaction between theparticles, anisotropy energy, an applied switching field with finitespatial extent and a small bias field are considered. When theseparation between the particles is small, the dipolar field issignificant and it affects the reversal of the magnetic moments. Thefinal configuration attained depends sensitively on the decaying lengthof the switching field, the inter-particle separation, and the initialconfiguration. A bias field tends to suppress the effects of aspatially decaying switching field and dipolar interaction betweenneighbouring particles.

3D Analogue Models of Rift Systems: Templates for 3D Seismic Interpretation

Abstract 3D visualizations of modern, high-resolution seismic data have provided valuable insights into the finite geometries and spatial extent of extensional fault systems, but their evolution in time is poorly understood. Scaled 3D analogue models of rift basin evolution provide kinematic templates for understanding the 4D evolution of extensional fault systems. This paper reviews the development of extensional fault systems in analogue models of orthogonal, oblique and offset rifts. In orthogonal and oblique models, stretching above a zone of ductile deformation at the base of the model initially produced segmented rift border faults whose orientations were strongly controlled by the underlying baseplate configuration. In contrast, the intra-rift faults generally initiated at high angles to the extension direction. With increased extension both the rift border faults and the intra-rift faults propagated along strike, first producing segmented fault systems separated by relay ramps, which, with increased extension, became breached as fault linkage occurred. Kinks in the fault traces indicate linkage points. Within the models, asymmetric intra-rift sub-basins were formed where the extensional fault arrays had a dominant dip polarity. Intra-basin accommodation zones, separating individual sub-basins along the rift axis, were formed by interlocking oppositely dipping fault systems. Offset oblique rift models, formed above a zone of ductile stretching with basement offsets, generated intra-basin accommodation zones whose orientation was controlled by the underlying basement fabric. The results of the analogue models can be directly compared with fault systems in the Northern Ethiopian rift system, with the accommodation zones in the Gulf of Suez, Egypt, with extensional fault arrays in Canyonlands, Utah, and with rift fault systems in the Gulf of Thailand and the southern North Sea.

The pion mass in finite volume

We determine the relative pion mass shift $M_\pi(L)/M_\pi-1$ due to the finite spatial extent $L$ of the box by means of two-flavor chiral perturbation theory and the one-particle L\"uscher formula. We use as input the expression for the infinite volume $\pi\pi$ forward scattering amplitude up to next-to-next-to-leading order and can therefore control the convergence of the chiral series. A comparison to the full leading order chiral expression for the pion mass in finite volume allows us to check the size of subleading terms in the large-$L$ expansion.

A finite element–boundary element method for advection–diffusion problems with variable advective fields and infinite domains

In this paper a hybrid, finite element–boundary element method which can be used to solve for particle advection–diffusion in infinite domains with variable advective fields is presented. In previous work either boundary element, finite element, or difference methods were used to solve for particle motion in advective–diffusive domains. These methods have a number of limitations. Due to the complexity of computing spatially dependent Green's functions, the boundary element method is limited to domains containing only constant advective fields, and due to their inherent formulations, finite element and finite difference methods are limited to only domains of finite spatial extent. Thus, finite element and finite difference methods are limited to finite space problems for which the boundary element method is not, and the boundary element method is limited to constant advection field problems for which finite element and finite difference methods are not. In this paper it is proposed to split the total domain into two sub-domains, and for each of these sub-domains, apply the appropriate solution method; thereby, producing a method for the total infinite space, variable advective field domain.

Quantification of molecular cloud structure using theΔ-variance

We present a detailed study of the Δ -variance as a method to quantify molecular cloud structure. The Δ -variance was introduced by Stutzki et al. (1998) to analyze the drift behaviour of scalar functions and is used to characterize the spatial structure of observed molecular cloud images. For fractional Brownian motion structures ( fBm-fractals ), characterized by a power law power spectrum and random phases, the Δ -variance allows to determine the power spectral index β . We present algorithms to determine the Δ -variance for discretely sampled maps and study the influence of white noise, beam smoothing and the finite spatial extent of the maps. We find that for images with , edge effects can bias the structure parameters when determined by means of a Fourier transform analysis. In contrast, the Δ -variance provides a reliable estimate for the spectral index β , if determined in the spatial domain. The effects of noise and beam smoothing are analytically represented in a leading order approximation. This allows to use the Δ -variance of observed maps even at scales where the influence of both effects becomes significant, allowing to derive the spectral index β over a wider range and thus more reliably than possible otherwise. The Δ -variance is applied to velocity integrated spectral line maps of several clouds observed in rotational transitions of 12 CO and 13 CO. We find that the spatial structure of the emission is well characterized by a power law power spectrum in all cases. For linear scales larger than ∼ 0.5 pc the spectral index is remarkably uniform for the different clouds and transitions observed (). Significantly larger values () are found for observations made with higher linear resolution toward the molecular cloud MCLD 123.5+24.9 in the Polaris Flare, indicating a smoother spatial structure of the emission at small scales (<0.5 pc).

The Density Structure of Highly Compact H [CSC]ii[/CSC] Regions

We report the density structure of the ultracompact (UC) H ii regions G35.2021.74, G9.6210.19-E, and G75.7810.34-H O. The density profiles are derived from radio continuum emission at wavelengths from 6 to 2 0.3 cm. In the case of G35.2021.74, a cometary UC H ii region with a core and a tail, the spectrum of the core varies as , implying that the density structure is . The emission from the tail has a flatter spectrum, 0.6 22 S / n n / r n e indicating that the density gradient is also negative but shallower. For the case of G9.62 10.19, which is an H ii region complex with several components, the spectrum of the region designated component E is , cor0.95 S / n n responding to . The steepest spectral index, , is for the super UC H ii region G75.7810.34-H O; 22.5 1.4 n / rS / n e n 2 its density stratification may be as steep as . The actual density gradient may be smaller, owing to an 24 n / r e exponential (rather than a power-law) density distribution or to the effects of finite spatial extent. The contribution from dust emission and some of the possible implications of these density distributions are briefly discussed. Stellar groups form in the dense cores of molecular clouds. The structure of these clouds has been unveiled with optical, infrared, and radio observations over the last three decades, and a great effort has been made to understand the connections between cloud properties and star-forming activity. In particular, knowledge of the gas density structure is required to determine fundamental properties, such as the mass and stability of star-forming cores. Extinction studies can be used to derive the density profiles of clouds, but given the large opacities involved, they can only probe the outermost gas layers. Tracers occurring at radio frequencies, on the other hand, can penetrate deeper into the cloud and can reveal the density stratification of dark clouds and cloud cores. The information obtained from extinction and molecular-line studies shows that molecular clouds are centrally condensed. For instance, moderate-density (10 3 ‐10 5 cm ) envelopes sur23 rounding higher density (10 7 cm ) cores suggest the existence 23

CMB anisotropy in compact hyperbolic universes. II. COBE maps and limits

The measurements of CMB anisotropy have opened up a window for probing the global topology of the universe on length scales comparable to, and even beyond, the Hubble radius. For compact topologies, the two main effects on the CMB are: (1) the breaking of statistical isotropy in characteristic patterns determined by the photon geodesic structure of the manifold and (2) an infrared cutoff in the power spectrum of perturbations imposed by the finite spatial extent. We calculate the CMB anisotropy in compact hyperbolic universe models using the regularized method of images described in detail in the preceding paper, including the line-of-sight ``integrated Sachs-Wolfe'' effect, as well as the last-scattering surface terms. We calculate the Bayesian probabilities for a selection of models by confronting our theoretical pixel--pixel temperature correlation functions with the COBE-DMR data. Our results demonstrate that strong constraints on compactness arise: if the universe is small compared to the horizon size, correlations appear in the maps that are irreconcilable with the observations. This conclusion is qualitatively insensitive to the matter content of the universe, in particular, the presence of cosmological constant. If the universe is of comparable size to the ``horizon,'' the likelihood function is very dependent upon orientation of the manifold w.r.t. the sky. While most orientations may be strongly ruled out, it sometimes happens that for a specific orientation the predicted correlation patterns are preferred over those for the conventional infinite models. The full Bayesian analysis we use is the most complete statistical test that can be done on the cosmic background explorer maps, taking into account all possible signals and their variances in the theoretical skies, in particular the high degree of anisotropic correlation that can exist. We also show that standard visual measures for comparing theoretical predictions with the data such as the isotropized power spectrum ${C}_{l}$ are not so useful in small compact spaces because of enhanced cosmic variance associated with the breakdown of statistical isotropy.

Ring-like vortex solutions to the Landau–Ginzburg equations in superconducting mesoscopic devices

Beyond the well-known Abrikosov and giant vortex configurations, we describe new types of solutions to the Landau–Ginzburg (LG) equations for low temperature superconductivity. These solutions correspond to ring-like current vortices, surrounding one another in concentric patterns, which in addition may possibly be bound to a standard L fluxoid configuration in their center. These solutions are possible in mesoscopic devices, due to their finite spatial extent, and generalize the usual vortex configurations for large enough samples. They contribute additional states at a given applied magnetic field, whose metastability still needs to be assessed. Such configurations could add to the richness of the phase transition behavior of mesoscopic superconducting disks and annuli in external magnetic fields, in ways to be explored. The properties and characterization of these new configurations are described, both for mesoscopic disks and annuli.

Large-scale vorticity generation by breakers in shallow and deep water

Water wave breaking is of considerable importance in the transfer of momentum, and in other transfers, between the atmosphere and oceans. Typically breaking occurs on deep water as events that have finite duration and finite spatial extent. Near shore lines most of the water motions are dominated by breaking waves. Recent work on the generation of vorticity by breaking waves and bores in the surf zone on beaches is considered and typical vortical structures are briefly discussed. Consideration of deep water breaking leads to the proposal that the end result of a breaking event in deep water may be a coherent structure within the resulting current field. Such a structure is topologically equivalent to half a vortex ring.

Computing CMB anisotropy in compact hyperbolic spaces

The measurements of CMB anisotropy have opened up a window for probing the global topology of the universe on length scales comparable to and beyond the Hubble radius. For compact topologies, the two main effects on the CMB are: (i) the breaking of statistical isotropy in characteristic patterns determined by the photon geodesic structure of the manifold and (ii) an infrared cut-off in the power spectrum of perturbations imposed by the finite spatial extent. We present a completely general scheme using the regularized method of images for calculating CMB anisotropy in models with non-trivial topology, and apply it to the computationally challenging compact hyperbolic topologies. This new technique eliminates the need for the difficult task of spatial eigenmode decomposition on these spaces. We estimate a Bayesian probability for a selection of models by confronting the theoretical pixel-pixel temperature correlation function with the COBE-DMR data. Our results demonstrate that strong constraints on compactness arise: if the universe is small compared to the `horizon' size, correlations appear in the maps that are irreconcilable with the observations. If the universe is of comparable size, the likelihood function is very dependent upon orientation of the manifold with respect to the sky. While most orientations may be strongly ruled out, it sometimes happens that for a specific orientation the predicted correlation patterns are preferred over the conventional infinite models.

Thermodynamics of the two-dimensional O(3) σ-model with fixed-point lattice action

The free energy density of the two-dimensional O(3) non-linear σ-model is calculated at finite temperature and finite spatial extent. Both an analytic calculation in the perturbative regime and a Monte Carlo study at low temperatures are presented. It is shown that using the fixed-point action instead of the standard Wilson action leads to a substantial reduction of the cut-off effects.

Gravitational radiation from compact binary systems: Gravitational waveforms and energy loss to second post-Newtonian order.

We derive the gravitational waveform and gravitational-wave energy flux generated by a binary star system of compact objects (neutron stars or black holes), accurate through second post-Newtonian order ($O[(v/c)^4] \sim O[(Gm/rc^2)^2]$) beyond the lowest-order quadrupole approximation. We cast the Einstein equations into the form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve it formally as a retarded integral over the past null cone of the chosen field point. The part of this integral that involves the matter sources and the near-zone gravitational field is evaluated in terms of multipole moments using standard techniques; the remainder of the retarded integral, extending over the radiation zone, is evaluated in a novel way. The result is a manifestly convergent and finite procedure for calculating gravitational radiation to arbitrary orders in a post-Newtonian expansion. Through second post-Newtonian order, the radiation is also shown to propagate toward the observer along true null rays of the asymptotically Schwarzschild spacetime, despite having been derived using flat spacetime wave equations. The method cures defects that plagued previous ``brute- force'' slow-motion approaches to the generation of gravitational radiation, and yields results that agree perfectly with those recently obtained by a mixed post-Minkowskian post-Newtonian method. We display explicit formulae for the gravitational waveform and the energy flux for two-body systems, both in arbitrary orbits and in circular orbits. In an appendix, we extend the formalism to bodies with finite spatial extent, and derive the spin corrections to the waveform and energy loss.

Nonlinear Mode Localization in Systems Governed by Partial Differential Equations

The concept of nonlinear normal mode (NNM) is used to study localized oscillations in certain classes of oscillators governed by nonlinear partial differential equations. NNMs are synchronous free oscillations during which all positional coordinates of the system reach their extreme values or pass through the equilibrium position at the same instant of time. Although such motions can be regarded as nonlinear analogs of the linear normal modes of classical vibration theory, not all NNMs are analytic continuations of linear ones. Continuous systems of finite and infinite spatial extent are considered. For periodic assemblies consisting of a finite number of nonlinear structural members, the NNMs are computed asymptotically by solving nonlinear sets of equations possessing regular singular points. Some of the computed NNMs are spatially localized to only a limited number of components of the assembly. The bifurcations giving rise to nonlinear mode localization are examined using the perturbation method of multiple-scales. The implications of nonlinear mode localization on the vibration and shock isolation of periodic flexible structures are discussed. In particular, localized NNMs lead to passive motion confinement of disturbances generated by impulsive loads. Finally, the concept of NNMs is extended to analytically study standing waves with spatially localized envelopes in a class of nonlinear partial differential equations defined over infinite domains. It is shown that NNM-based methodologies can be an effective tool for analyzing such motions.

Elastic waves in homogeneous and layered transversely isotropic media: Plane waves and Gaussian wave packets. A general approach

Solutions to the equation of motion are derived for transversely isotropic media such as fiber composites, ideally fiber-textured austenitic steels, or extruded metal-matrix composites. The approach is most general in that the orientation of the materials’ axis of rotational symmetry is arbitrary. Thus the results obtained using a coordinate-free representation are particularly convenient in view of layered structures, where for the materials of interest the fiber axis is perpendicular to the surface normal, but variable in orientation. Plane elastic waves are characterized by the corresponding wave vectors, making especially possible a quantitative evaluation of the deviation of wave propagation direction and energy flux, which is characteristic for anisotropic materials. Reflection and refraction of plane waves at an interface between two arbitrarily oriented transversely isotropic media is examined yielding an algorithm that provides the respective reflection and transmission coefficients. The propagation of elastic waves of finite spatial and temporal extent is modeled using the concept of Gaussian wave packets. The relations given in the literature for general anisotropic media are specialized to the homogeneous and layered transversely isotropic cases. Numerical evaluation of the analytical results is included.

Time-dependent inviscid vortices in global self-similar expansion

A family of time-dependent solutions is derived, which describe the self-similar motion of an inviscid γ=5/3 adiabatic fluid in arbitrary three-dimensional geometry. The derivation identifies a non-Galilean frame of reference in which the self-similar flows are observed to be steady, subject to a (fictitious) inertial force. This theory generalizes a primitive version, previously published, to allow for quite general velocity structures which are topologically no less complex than those belonging to the set of all steady flows. The existence of these self-similar flows stems from a compatibility between the conservation laws governing the internal energy of the γ=5/3 adiabatic gas and its angular momentum. Illustrative analytic solutions are presented for vortices of finite spatial extent isolated in a body of fluid undergoing purely radial expansion. These vortices consist of axisymmetric, closed meridional circulations, with or without swirls. A related set of self-similar, magnetohydrodynamic solutions for the γ=2 adiabatic fluid in the Cartesian plane are derived in the second of two appendices.

Guided wave propagation across sharp lateral heterogeneities: the complete wavefield at a cylindrical inclusion

We present an exact treatment of wave propagation across a cylindrical inclusion embedded in a layered waveguide, which may serve as a test case allowing an estimation of the accuracy and validity range of approximation methods applied to surface wave propagation. Since, in contrast to a plane vertical interface, a cylindrical inclusion is of finite spatial extent, the test case presented here is especially well suited to assess the quality of approaches based on scattering theory. The wavefield is represented by Love and Rayleigh type modes. A 3-D orthonormality relation is derived, expressed as an integral over the cylinder surface which allows a direct and unique computation of basis solutions. After solving a linear system of equations these basis solutions are superimposed to form the complete wavefield both within and outside the cylinder. It is shown that exact continuity of displacements and tractions at the interface can not be achieved with propagating modes only. Non-propagating modes, which have complex wavenumbers and hence decay in the propagation direction, have to be included in the modal series. In contrast to propagating modes, complex modes have vanishing energy flux. Numerical results are presented for a layered halfspace, where only propagating modes were used, and for a layered waveguide for various sizes of the cylindrical inclusion. Astonishingly, vertical and horizontal displacements behave very differently. For instance, the scattered vertical displacement generated by an incoming Rayleigh fundamental mode is clearly dominated by the mode itself, while the scattered horizontal displacement is severely influenced by the excited higher Rayleigh modes. This might explain the difficulties met when interpreting horizontal components of surface wave data within a single-mode concept.

Noncollocated point control of nondispersive distributed-parameter systems using time delays

This paper deals with the noncollocated, time-delayed active point control of continuous systems. It considers systems of finite spatial extent which can be modeled by the undamped wave equation. The paper presents a new method of control of the system using noncollocated sensors and actuators. By using the physical properties of the mode shapes of vibration of the system it is shown that the modal response at a given location in the system can be reconstructed from the time-delayed modal responses at (at most) three different locations in the system. This result is then used to motivate a closed-loop control design which is capable of stabilizing the system and dampening the vibrations in all its modes, using dislocated sensor and actuator locations. It is found that the cost of dislocating the sensors from the actuator is at most an additional two sensors. For special types of boundary conditions this cost may be reduced to one additional sensor, or even be completely eliminated. Simple finite-dimensional controllers, commonly used in control design, are found to suffice. The results are valid for rather general conditions at the boundaries of the continuum. Explicit conditions are provided to obtain the bounds on the controller gains to ensure stability of the closed-loop control design. These bounds are obtained in terms of the locations of the sensors and the actuators. Simulation results, which validate the control methodology and the theoretical bounds on the gain, are also presented.

Effect of spatial and velocity distributions on the generation of ion cyclotron emission from fusion products

Cyclotron motion of fusion products (FPs) or other superthermal ions has been shown theoretically to generate ion cyclotron emission (ICE) which is strongly dependent upon the ratio of ion velocity to the background Alfvén velocity [C. Chu and J. L. Sperling, J. Plasma Phys. 40, 40 (1988)]. This initial work described ICE assuming a uniform plasma embedded in a constant magnetic field; in addition, a delta function velocity distribution was used for the superthermal ions. These assumptions lead to ICE spectra which exist as a series of delta functions at harmonics of the fundamental cyclotron frequency of the superthermal species. It is shown here that source distributions with finite spatial extent such as a higher order parabolic model ((S(R)=S0{1−[(R−R0) /a]2}N,N=6–10)), in conjunction with the magnetic field profile ((B(R)=B0R0 /R)), lead to a significant broadening of the ICE spectra; this is the case even for a delta function velocity distribution. The spectra are further modified when realistic superthermal distributions are employed in the model. In the latter case, a Green’s function approach is employed to calculate the spectra. Results of the ICE calculations are presented for FP protons.

Influence of a finite scattering volume on the determination of electron-impact coherence parameters.

Superelastic electron scattering from laser-excited Ba-138 ...6s6p (1)P is investigated both theoretically and experimentally. The theoretical framework is developed and incorporated into a model of a realistic scattering geometry of finite spatial extent. Model calculations predict that the dependence of the superelastic scattering intensity on the linear polarization direction of the laser beam can be severely distorted due to the existence of a finite scattering volume. Measurements obtained confirm the presence of this distortion and are well described by the model calculations. Conclusions are drawn concerning the influence of this effect on the extraction of electron-impact coherence parameters.