Two-dimensional Version Research Articles

Two-dimensional Sen connections in general relativity

The two-dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric on the spin spaces is found which characterizes both the algebraic and extrinsic geometrical properties of the 2-surface . The curvature of the two-dimensional Sen operator is the pullback to of the anti-self-dual part of the spacetime curvature, while its `torsion' is a boost-gauge invariant expression of the extrinsic curvatures of . The difference between the two-dimensional Sen and the induced spin connections is the anti-self-dual part of the `torsion'. The irreducible parts of are shown to be the familiar 2-surface twistor and the Weyl--Sen--Witten operators. Two Sen--Witten type identities are derived; the first is an identity between the two-dimensional twistor and the Weyl--Sen--Witten operators and the integrand of Penrose's charge integral, while the second contains the `torsion' as well. For spinor fields satisfying the 2-surface twistor equation the first reduces to Tod's formula for the kinematical twistor.

Current Advance Method and Cyclic Leapfrog for 2D Multispecies Hybrid Plasma Simulations

CAM-CL (current advance method and cyclic leapfrog) is a new algorithm for hybrid plasma simulations. In common with existing methods, its physical basis is a "hybrid" plasma model which treats the ions as particles and the electrons as a massless fluid. CAM-CL is distinguished from previous 2D hybrid algorithms by four main features: (1) Multiple ion species may be treated with only a single computational pass through the particle data: this is achieved without extrapolation of the electric field in time. The particles are advanced by a leapfrog procedure which requires the electric field to be a half time-step ahead of the particle velocities. The electric field depends on the ionic current density and hence the particle velocities. In order to avoid a time-consuming "pre-push" of the velocities, CAM advances the ionic current density a half-step with an appropriate equation of motion. This is similar in concept to the moment method (D. Winske and K. B. Quest, J. Geophys, Res.93 (A9), 9681 (1988), Appendix A), except for the next two features: (2) CAM advances the ionic current density, whereas the moment method advances the fluid velocity. Consequently, multiple ion species may be easily treated (3) A free-streaming ionic current density is collected (velocities are collected at positions a half time-step ahead). An equation of motion is then applied, in which the advective term and the ionic stress tensor in the moment method are not needed, since transport effects are included in the free-streaming current. (4) CL is a leapfrog scheme for advancing the magnetic field, an adaptation of the modified midpoint method described by W. H. Press et al. (Numerical Recipes (Cambridge Univ. Press, Cambridge, 1986)). It is stable and allows sub-stepping of the magnetic field (the magnetic field time-step may be different to the particle time-step). A two-dimensional version of the algorithm has been tested on a quiet plasma, MHD wave propagation, and ion beam instabilities, the results of which are discussed.

Assignment of NMR spectra of proteins using triple-resonance two-dimensional experiments

Two-dimensional versions of HNCA and HNCO experiments are described, which provide essentially the same information as the 3D sequence. A multiple-quantum coherence involving either 15N and 13C alpha or 15N and 13CO is created. One of the two frequencies is given by the middle point between the two cross peaks (zero- and double-quantum) and the other by their separation. Quadrature detection can be performed on either nucleus, modifying only the appearance of the 2D spectrum, but not the information content. These experiments, named MQ-HNCA and MQ-HNCO, are illustrated on a (15N, 13C) doubly labelled cytochrome c2 from Rhodobacter capsulatus (116 amino acids).

A MUSIC-like method for estimating quadratic phase coupling

In this paper, a two-dimensional version of the well-known MUSIC algorithm for estimating the quadratically coupled frequency pairs (QC pairs) in a noise-corrupted complex harmonic process is proposed. It is shown that the algorithm can also, with minor modifications, be used for estimating the bispectrum of a general third-order stationary harmonic process. The algorithm involves arranging the complex third-order cumulants of the noisy harmonic process in the form of a matrix having a normal structure, i.e., a matrix with an orthonormal eigenbasis. It is shown that a necessary and sufficient condition for an ordered pair ( θ 1 , θ 2 ) in the two-dimensional frequency plane to be a QC pair is that the Kronecker product between steering vectors associated with the two frequencies θ 1 and θ 2 lies in the signal subspace of this matrix. By exploiting this result and the orthogonality between the signal and noise subspaces of the above matrix, a symmetric search function of two frequency variables (termed as the MUSIC pseudo-bispectrum estimator) is constructed using the signal eigenvectors. This function is shown to peak precisely at the QC pair locations in the two-dimensional frequency plane. Simulation results are presented, and are shown to testify to the high-resolution performance of this estimator. Eine zweidimensionale Version des bekannten MUSIC-Algorithmus' zur Schätzung quadratisch gekoppelter Frequenzpaare (QC-Paare) in einem verrauschten komplexen harmonischen Prozess wird vorgeschlagen. Es zeigt sich, dass man das Verfahren mit geringfügigen Modifikationen auch zur Schätzung des Bispektrums eines allgemeinen stationären harmonischen Prozesses dritter Ordung verwenden kann. Der Algorithmus beinhaltet die Anordnung der komplexen Kumulanten dritter Ordnung des verrauschten harmonischen Prozesses in Matrixform mit Normalstruktur, d.h. in einer Matrix mit orthonormaler Eigenbasis. Es wird gezeigt, dass eine notwendige und hinreichende Bedingung dafür, dass ein geordnetes Paar ( θ 1 , θ 2 ) in der zweidimensionalen Frequenzebene ein QC-Paar darstellt, darin besteht, dass das Kroneckerproduckt der Steuervektoren zu den beiden Frequenzen θ 1 und θ 2 im Signal-Unterraum der Matrix liegt. Indem dieses Resultat und die Orthogonalität zwischen Signal- und Rauschunterraum der Matrix ausgenützt werden, lässt sich eine symmetrische Suchfunktion der beiden Frequenzvariablen (als MUSIC-Pseudobispektrums-Schätzer bezeichnet) unter Verwendung der Signal-Eigenvektoren konstruieren. Es wird gezeigt, dass diese Funktion ihre Maxima exakt in den Punkten des QC-Paares in der komplexen Frequenzebene besitzt. Simulationsergebnisse werden vorgestellt. Sie belegen die Leistungsfähigkeit und die hohe Auflösung des Schätzers. Nous proposons dans cet article une version bidimensionnelle de l'algorithme bein connu MUSIC pour l'estimation de paires de fréquence en couplage quadratique (paires QC) dans un processus harmonique complexe bruité. Nous montrons que l'algorithme peut également, au prix de modifications mineures, être utilisé pour estimer le bispectre d'un processus harmonique stationnaire du troisième ordre.général. Cet algorithme requièrt l'arrangement des cumulants complexes d'ordre trois du processus harmonique bruité sous la forme d'une matrice ayant une structure normale, à savoir une matrice ayant des vecteurs propres orthonormaux. Nous montrons qu'une condition nécessaire et suffisante pour qu'une paire ordonnée ( θ 1 , θ 2 ) dans le plan fréquentiel bidimensionnel soit une paire QC est que le produit de Kronecker entre les vecteurs associés aux deux fréquences θ 1 et θ 2 se trouve dans le sous-espace de signal de la matrice. Nous appuyant sur ce résultat et sur l'orthogonalité des sous-espaces de signal et de bruit de la matrice mentionnée ci-dessus, nous construisons une fonction de recherche symétrique des deux variables de fréquence (appelée estimateur MUSIC du pseudo bispectre) à l'aide des vecteurs propres correspondant au signal. Nous montrons que cette fonction possède un pic situé précisément sur les paires QC dans le plan fréquence bidimensionnel. Nous présentons des résultats de simulation qui mettent en évidence les capacités de haute résolution de cet estimateur.

Shell dynamics with three-dimensional degenerate finite elements

An explicitly through the thickness integrated two-dimensional version of the three-dimensional degenerated shell element is formulated here to study the dynamics of elastic shells. A nine-noded quadrilateral Lagrangian element is used with five degrees of freedom per node. A specialized mass diagonalization scheme, developed by Hinton, Rock and Zienkiewicz, is used which conserves the total mass of the element and also includes the effects of the rotary inertia terms. Hamilton's principle is used to derive the equations of motion. Mode superposition coupled with Duhamel's integral is first employed to obtain a solution of the equations of motion in time. Mode shapes and frequencies are computed by subspace iteration technique. Direct time integration using the implicit Newmark-β method is also carried out. Several examples are presented and the results obtained by mode superposition and direct time integration methods are compared.

Modified forms of the third-order convection, second-order diffusion scheme for the advection-diffusion equation

This paper discusses and compares the spatial accuracy of the QUICK finite difference scheme and the third-order convection, second-order diffusion (TCSD) scheme for the severe case with pure advection only. It is shown that the QUICK scheme is generally second-order accurate in space and that a general explicit finite difference representation of various upwind difference schemes is numerically unstable for the severest transport case. Various modified forms of the implicit TCSD scheme are presented with their numerical stability properties being studied and analysed. These modified schemes have been applied to an idealised one-dimensional test basin for the cases of pure advection and of advection and diffusion using three different initial-boundary conditions, including: (a) a sharp front concentration gradient; (b) a Gaussian concentration distribution; and (c) a plug source. A two-dimensional version of the modified TCSD scheme has also been formulated based upon the standard ADI technique and has been applied to three two-dimensional test cases, including pure advection for a circular column and a Gaussian distribution, and advection with diffusion for a Gaussian distribution. Both uniform and rotational flow fields were used for these two-dimensional tests. The scheme has been shown to be attractive, with details of the comparisons between these modified schemes and other similar higher order schemes together with the corresponding analytical solutions also being included in the paper.

A new version of the two-dimensional Lax-Friedrichs scheme

We develop a new two-dimensional version of the Lax-Friedrichs scheme, which corresponds exactly to a transport projection method. The scheme we obtain in this way is different from the one derived by averaging the one-dimensional scheme in the two directions as usually done. The Lax-Friedrichs scheme is known to be a very stable scheme with much diffusion. However, this diffusion can be easily reduced by using corrected fluxes, without altering the total-variation estimates. The accuracy of this corrected scheme is of order two except near a local extrema. The numerical results computed by using this corrected scheme are similar to the ones obtained by using the Godunov scheme with corrected fluxes but require less CPU time. Convergence towards the entropy solution is proved, and some extensions to systems of conservation laws or three-dimensional models are discussed. Some numerical experiments are reported.

A numerical study of boundary-layer dynamics in a mountain valley

A higher order closure model is applied to simulate the dynamics in an area with a deep valley characterized by complex terrain in the southwestern US. The simulation results show generally good agreement with measured profiles at two locations within the valley. Both the measurements and the simulations indicate that the flow dynamics in the area are highly influenced by the topography and meandering of the valley, and can be resolved only by the full three-dimensional model code. The wind veering simulated over the range of the topographic elevations is often larger than 100 deg and in some cases as large as 180 deg, as a consequence of topographic forcing. In the case of an infinitely long valley, as is assumed in two-dimensional test simulations, a strong low-level jet occurs within the valley during stable conditions. The jet is mainly a consequence of the Coriolis effect. However, the jet development is significantly reduced due to asymmetric effects of the actual topography treated in the three-dimensional simulations. Tests with the two-dimensional nonhydrostatic version of the model show significant wave responses for a stable stratified flow over the valley. The structure resembles nonlinear mesoscale lee waves, which are intrinsically nonhydrostatic. However, considering the three-dimensional nature of this valley system, a better understanding and verification of the nonhydrostatic effects requires both a three-dimensional nonhydrostatic numerical model and an observational data set which is fully representative in all three dimensions.

Simulation and Animation of Musculoskeletal Joint System

This paper describes the development of computer-based software for three-dimensional geometric data base of the human musculoskeletal system. Using a computer graphics workstation, a user of the software will interactively display detailed information about the muscles, tendons, ligaments, bone, and joint anatomy. This software will enable a wide range of health care workers to visualize complex physiological data. In addition to geometric and visual realism, this software will include kinematic relationships which allow the calculation and display of the motion and forces of the joints, muscles, and tendons. This will permit a user to interactively move joints or tendons and display the resulting motion of the surrounding tissues, as well as internal reactive forces and joint pressure distribution. A two-dimensional version of this software is currently being used for knee and hip osteotomy preoperative planning, total joint replacement prosthesis design and dimensional selection, and osteochondral allograft sizing and reconstruction using radiographic data.

A nonhydrostatic model for simulation of airflow over mesoscale bell-shaped ridges

This paper describes a nonhydrostatic and incompressible mesoscale model formulation using a terrain-following coordinate system. A tensor transformation procedure is used to derive a diagnostic equation for the nonhydrostatic pressure field. The model features a simplified second-order turbulence closure scheme. The two-dimensional version of the nonhydrostatic model, as well as the corresponding hydrostatic model, are applied to simulate stably stratified airflow over mesoscale bell-shaped mountain ridges. The results show that the nonhydrostatic model is capable of simulating nonhydrostatic dynamics of mesoscale lee wave systems such as the trapped wave phenomenon.

Diffusion of CO 2 to the atmosphere from a source in the deep ocean

A two-dimensional version of the classical diffusion equation has been used to study the spreading of CO 2 from a source in the deep ocean. In the upper part of the ocean there is a mixed layer including full carbon chemistry, communicating with the atmosphere through a standard gas exchange formulation. The size of the vertical diffusion coefficient, the total ocean depth, the injection depth and the mixed layer thickness appear to be critical for the amount of CO 2 that outgasses to the atmosphere. The carbon source has been fitted to the emission of a 2 GW gas power plant operating for 100 years. If this amount of carbon is released at a depth of 2000 m, the cumulative outgassing to the atmosphere 200 years after the end of the operation of the plant is estimated to be 4% of the total injected carbon for average north Atlantic and north Pacific waters, and 14% for the Norwegian Sea.

Ground state of monolayer 4He/graphite: Quantum liquid versus quantum solid.

Two extensions are made to calculations which represent monolayer $^{4}\mathrm{He}$/graphite as a two-dimensional (2D) version of 3D helium: inclusion of substrate-mediated interactions and of modulation of the liquid phase by the substrate corrugation potential. A variation-perturbation approximation is developed for the modulation energy. The McLachlan energy is a substantial correction to the 2D energy calculation, while the energy of elastic distortion of the substrate is a small term. Consequences for the nature of the ground state of $^{4}\mathrm{He}$/graphite are discussed. If the McLachlan energy is a fair approximation to the energy shift by adsorption-induced processes and the substrate corrugation has, or is somewhat less than, the value used in recent work, the calculated ground state is a quantum liquid.

The role of anomalous singularity in hadronic form factors

By assuming chiral-symmetry breaking as the dominant dynamics in the light-hadron structure, we study the low-energy electromagnetic form factor of light hadrons in the Nambu-Jona-Lasinio type of models in the large- N limit from a dispersion relation point of view. In addition to the expected vector-meson pole dominance picture for tightly bound states such as the pion, it is found that the anomalous singularity, occuring only to loosely bound states of the constituent quarks, could severely alter the form factors at low momentum-transfers from the vector-meson dominance behavior. This mechanism is explicitly demonstrated in a two-dimensional version of the NJL type models, the Gross-Neveu model, in which all the interested quantities are calculated analytically and a detailed comparison with two-dimensional QCD is made. The two-dimensional analysis is then argued to be qualitatively similar to the realistic four-dimensional case. The relevance of the anomalous singularity to the phenomenology of the rho and the nucleons is indicated by quantitative estimates of the anomalous thresholds in their form factors.

Nonlinear Hydrostatic Conditional Symmetric Instability: Implications for Numerical Weather Prediction

A simplified two-dimensional version of the Pennsylvania State University-NCAR Mm scale Model (MM4) was used to investigate the nonlinear evolution of unforced conditional symmetric instability (CSI). Sensitivity tests examines the effects of the model resolution, the magnitudes and formulation of the vertical and horizontal diffusion, and the Prandtl number on various aspects of the CSI circulations. Aspects of the circulations that are considered include the transverse velocity, the growth rate, and the updraft slope, as well as the existence of a CSI circulation. With the fairly unstable initial conditions used in these tests, unstable CSI modes are explicitly resolved with horizontal grid spacing of at most 30 km, using a fairly weak horizontal diffusion. The vertical grid spacing needs to be no greater than 340 m to be consistent with the horizontal grid spacing and the sloping thermal structures. To resolve the most unstable nonlinear CSI modes, horizontal resolutions of no more than 15 km and vertical resolutions less than 170 m are necessary. The magnitude of the horizontal diffusion, the magnitude and the formulation of the vertical diffusion, and the Prandtl number all affect the simulated CSI circulations. Comparisons to linear theory allow the generalization of the results. These generalizations suggest that the combined effects of model resolution, model diffusion, scale height of the instability, and environmental conditions (e.g., N2) determine whether CSI will be explicitly released in a numerical model, and also whether the resolved unstable modes include the desirable most unstable mode. Unstable CSI modes that do not correspond to the most unstable mode can be explicitly resolved with a grid spacing that is coarser than that necessary for the most unstable mode. If the resolved modes do not include the most unstable mode, the evolution of the CSI circulations will be substantially slower and the amplitudes reduced from what should be expected in the real atmosphere in which the most unstable mode should be present.

Fractal energy surface structure of coincident-site lattice boundaries

Abstract The continued-fraction expansion algorithm is utilized to clarify the hierarchy among the coincident-site lattice structures. Boundary energies are calculated against twist misorientation based on a two-dimensional version of the model suggested by du Plessis and van der Merwe in 1965, which reveal a fractal energy surface with cusps at very σ value. The origin of the cusps and the effects of local structural relaxation on the boundary energies are discussed.

The cosmic radiation in the heliosphere at successive solar minima: Steady state no‐drift solutions of the transport equation

Cosmic ray intensities and density gradients, as observed by the Pioneer 10 and 11, Voyager 1 and 2, and IMP spacecraft during the 1977 and 1987 solar minimum periods are interpreted. This is done with a two‐dimensional, no‐drift version of the cosmic ray transport equation. While this oversimplified model obviously cannot fit all the observations satisfactorily, it does produce useful insight into spectral shapes in the outer heliosphere, the question of anomalous hydrogen, and the properties of radial gradients. These effects are discussed systematically, and it is shown that many of them are due to adiabatic losses inherent in the modulation process. It is also pointed out which effects need more sophisticated models, including drifts, time‐dependent effects and acceleration at the termination shock. In particular, it is demonstrated that there is probably no physically realistic way to reproduce the observed negative latitudinal gradients without resorting to drift models of the modulation. The best fit to the anomalous oxygen data is achieved with either a charge state of 2+, or with a source function that changes between solar minimum periods with a factor of 2.8.

Improved dynamic model of the human knee joint and its response to impact loading on the lower leg.

Almost a decade ago, three-dimensional formulation for the dynamic modeling of an articulating human joint was introduced. Two-dimensional version of this formulation was subsequently applied to the knee joint. However, because of the iterative nature of the solution technique, this model cannot handle impact conditions. In this paper, alternative solution methods are introduced which enable investigation of the response of the human knee to impact loading on the lower leg via an anatomically based model. In addition, the classical impact theory is applied to the same model and a closed-form solution is obtained. The shortcomings of the classical impact theory as applied to the impact problem of the knee joint are delineated.

Note on time reversible chaos in mixmaster dynamics.

The Belinskii-Khalatnikov-Lifshitz (BKL) parametrization of the diagonal spatially homogeneous Bianchi IX vacuum (mixmaster) cosmology can be expressed as a discrete realization of a chaotic map. A well-known invertible two-dimensional version of their parametrization embodies the time reversibility of the model's Einstein equations and can be written (in both an epoch-era and era-era version) to emphasize the fact that the function that maps the first parameter is the inverse of that which maps the second. It is shown here that this can be understood in terms of the self-similar equilateral triangle minisuperspace equipotentials that describe the approximate dynamics in the anisotropy plane of minisuperspace. A new parameter is identified which has the same value in all the epochs of a given era. The BKL parameters of the two-dimensional map are different ratios of triangle scales to the new parameter. It is shown how one parameter controls the change of era (and thus the chaos) in the collapse direction while the other plays the same role for expansion.

Anderson localization in the time domain: Numerical studies of waves in two-dimensional disordered media

Experimental work in the time domain on Anderson localization and anomalous diffusion in dimensions greater than one is scant, being confined to some ultrasonic work by Weaver and, inasmuch as measurements of frequency–frequency intensity correlation functions imply measurements of time domain behavior, some microwave work by Genack and co-workers. Numerical work in the time domain is similarly poorly developed. No studies have been conducted with sufficient detail for determination of the effective transport behavior. This paper presents results obtained from new numerical experiments in an attempt to rectify this lack. A numerical model is introduced that allows the tracking of evolving wave energy density in a two-dimensional linear classical wave equation version of an Anderson model diagonally disordered Hamiltonian. The behavior of the evolving average wave energy density in effectively infinite large samples of this system (typically 300 sites by 100 sites) excited by tone burst line sources is presented and the results shown to collapse to an apparent single universal response function with interesting asymptotic character like exp(−r/λ−r2/4 β t) where r is the distance from the source, λ is the localization length, and β is a constant not equal to the bare diffusivity. The effect of dissipation is also investigated, and, consistent with previous demonstrations, found to be trivial. [Work supported by NSF MSS-91-14360.]

The 12/13 January 1988 Narrow Cold-Frontal Rainband Observed during MFDP/FRONTS 87. Part II: Microphysics

The microphysics of a narrow cold-frontal rainband (NCFR) observed during the MFDP/FRONTS87 experiment is investigated by using a microphysical retrieval model. The equations of evolution of the water substance and of the temperature are solved using a wind field prescribed from dual-Doppler radar observations. Different runs of the model were performed to investigate the role of various microphysical processes. All of them use a two-dimensional version of the model and give a solution for the steady state corresponding to the input wind field. The validity of this approach was checked a posteriori by comparing the results obtained from vertical cross sections at two different locations and two different times. In each case, the consistency of the results was controlled through comparisons with in situ measurements (aircraft, ground stations, and radiosondes) and radar reflectivity observations. The main result obtained from this study was that the precipitation associated with the NCFR was mostly composed of graupel particles, essentially formed by riming. Rain was produced by accretion of cloud water in the condensation zone and by melting of graupel. The choice of the type of ice-precipitating particles introduced in the model appeared very important. Only rimed particles (graupel) could reproduce observed precipitation. The precipitation efficiency was rather high (73%). The zone of light precipitation in which the NCFR was embedded seemed to play no-seeder role in the growth of precipitation in the NCFR, probably due to the overturning airflow located in the prefrontal zone. Another important result concerns the role of the microphysical processes on the thermodynamics. The temperature drop observed at low levels just behind the frontal discontinuity could be explained at the time of the observations by two cooling effects of equal importance: the melting of graupel and the evaporation of precipitation.