The shallow water equations in Lagrangian coordinates

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge–Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations.     

The single photon superradiance from the eigenmode analysis

Using the eigenmode analysis of the scalar photon theory, I compute the probability of the atoms remaining excited and the probability for the atoms remaining in the initial quantum state of a system of two-level atoms cloud in a sphere initially prepared to radiate in the forward direction, i.e., the single photon superradiance problem. The convergence in the results obtained for increasingly larger radii for the sphere suggests that the asymptotic limits for these quantities are obtained for a sphere with a radius equal to six times the resonant wavelength. I predict the maximal value of the probability of secondary excited states from large spheres at 17.1%.     

颗粒物质中的奇异现象

通过分析颗粒物质中一些有别于气体、液体和固体的奇异现象,结合生活和生产中的实际例子进一步认识颗粒物质的一些基本特性、规律、有待解决的问题及研究意义.     

A variant on eigenmode method in periodic crossed gratings

The eigenvalue problem discussed in Noponen and Turunen [Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles, J. Opt. Soc. Am. A 11 (1994) 2494–2502] for crossed gratings is analyzed in a different way using the three components of the electric field instead of the components E_x,y, H_x,y. As a result, half as many eigenvectors are needed.     

The strange behavior of entangled photons

An experiment is discussed in which a conflict between the wave like and nonlocal properties of photons leads to an apparent paradox. Possible ways of circumventing this contradiction are suggested. The experiment allows us to prove a Bell’s theorem for two particles without using an inequality and to test local realism against quantum mechanics introducing additional assumptions weaker than nonenhancement.     

The strange man in random networks of automata

We have performed computer simulations of Kauffman’s automata on several graphs, such as the regular square lattice and invasion percolation clusters, in order to investigate phase transitions, radial distributions of the mean total damage (dynamical exponent) and propagation speeds of the damage when one adds a damaging agent, nicknamed “strange man”. Despite the increase in the damaging efficiency, we have not observed any appreciable change of the transition threshold to chaos neither for the short-range nor for the small-world case on the square lattices when the strange man is added, in comparison to when small initial damages are inserted in the system. Particularly, we have checked the damage spreading when some connections are removed on the square lattice and when one considers special invasion percolation clusters (high boundary-saturation clusters). It is seen that the propagation speed in these systems is quite sensible to the degree of dilution on the square lattice and to the degree of saturation on invasion percolation clusters.     

The structure of a strange attractor

A few examples are given of an iterative map of the horseshoe type for which everything can be expressed in elementary functions. The strange attractor can be an analytic curve or a set of parallel lines with a Cantor set as its cross-section.     

The NUT metric in synchronous coordinates

The NUT metric is transformed to a synchronous frame with the help of a simple straightforward method. The three-space of this frame shows no singularity in the rate-of-strain tensor (acceleration and rotation being absent by definition) and in the three-curvature. Circular orbits with θ = const ≠ π/2 are also discussed.     

The molecular Hamiltonian in internal coordinates

A systematic approach for the derivation of the exact translational–rovibronic (non-relativistic) Hamiltonian for a polyatomic molecule consisting of N nuclei and n electrons is presented. All coupling terms which contribute to the total energy are identified. The Hamiltonian is greatly simplified by taking the internal coordinates (bond lengths and bond angles) as the vibrational variables. The translational–rovibronic Hamiltonian of triatomic molecules are considered as an application for this general formulation.     

The strange backbending behaviour in the Yb isotopes

The rotational energies and the backbending behaviour of ^166,168,170Yb are calculated utilizing angular momentum and particle number projected deformed BCS trial wave functions. The anomaly of the moment of inertia can be reproduced quantitatively with the moment of inertia of an inert core as the only free parameter. A possible explanation for the strange behaviour in the Yb isotopes is given.     

Advanced boundary conditions for eigenmode expansion models

In order to realise the full potential of eigenmode expansion models, advanced boundary conditions are required that can absorb the radiation impinging on the walls of the discretisation volume. In this paper, we will discuss and compare a number of these boundary conditions, like perfectly matched layers (PMLs), open (leaky mode) boundary conditions and transparent boundary conditions (TBCs). We will also introduce the case of PMLs with infinite absorption and discuss its relation to leaky mode expansion, leading to a deeper insight into the physics of PML.     

Advanced boundary conditions for eigenmode expansion models

In order to realise the full potential of eigenmode expansion models, advanced boundary conditions are required that can absorb the radiation impinging on the walls of the discretisation volume. In this paper, we will discuss and compare a number of these boundary conditions, like perfectly matched layers (PMLs), open (leaky mode) boundary conditions and transparent boundary conditions (TBCs). We will also introduce the case of PMLs with infinite absorption and discuss its relation to leaky mode expansion, leading to a deeper insight into the physics of PML.     

The strange border of the QCD phases

We address the flavour composition along the border between the hadronic and the quark–gluon plasma phases of QCD. The ratio of strange to up and down antiquarks () produced in particle and nuclear collisions is found to increase in collisions with an initially reached energy density () up to GeV/fm. Above this value it decreases approximately linearly and reaches its asymptotic value at zero baryon chemical potential (). We demonstrate that in nuclear collisions approaches its asymptotic value at –9 GeV/fm, corresponding to –8 TeV per pair, which will be reached at the LHC. After correcting for the difference in the chemical potentials of various colliding systems, universally saturates across the QCD phase boundary, following the temperature. Recent experimental puzzles as the increase in the ratio in collisions at 40 GeV per nucleon, its different behaviour at midrapidity, the decrease of the double ratio of in nucleus–nucleus over collisions with increasing , and the increase of in over collisions at the same , are naturally explained. We study the approach of thermodynamic observables at to the transition point and extract an estimate of the critical temperature. Received: 17 April 2001 / Published online: 24 August 2001     

The simplest case of a strange attractor

It is shown that stochastic motion of strange attractor type may arise in a system with stable limit cycle if the perturbation of the system is periodical. Analytical and numerical analyses of the conditions for the strange attractor are developed.     

Two-dimensional plasmonic eigenmode nanolocalization in an inhomogeneous metal-dielectric-metal slot waveguide

We show that transverse, with respect to the propagation direction, local narrowing of a metal-dielectricmetal plasmonic slot waveguide leads to a two-dimensional surface plasmon nanolocalization and can squeeze the plasmon eigenmode into a spot with a characteristic size of about several tens of nanometers. We demonstrate that the simultaneous waveguide tapering and decreasing transverse narrowing scale make possible an enhancement of the plasmon propagation distance in comparison with the homogeneous waveguide. We also find the fundamental limit of 2D plasmon nanolocalization, which is of the same order as the depth of penetration of the electromagnetic field into a metal which is actually independent of the field frequency in the near infrared domain.     

The combined Lagrangian advection method

We present and test a new hybrid numerical method for simulating layerwise-two-dimensional geophysical flows. The method radically extends the original Contour-Advective Semi-Lagrangian (CASL) algorithm [5] by combining three computational elements for the advection of general tracers (e.g. potential vorticity, water vapor, etc.): (1) a pseudo-spectral method for large scales, (2) Lagrangian contours for intermediate to small scales, and (3) Lagrangian particles for the representation of general forcing and dissipation. The pseudo-spectral method is both efficient and highly accurate at large scales, while contour advection is efficient and accurate at small scales, allowing one to simulate extremely fine-scale structure well below the basic grid scale used to represent the velocity field. The particles allow one to efficiently incorporate general forcing and dissipation.     

托卡马克中漂移不稳定性的回旋动理学二维本征模方程

从弗拉索夫方程出发,导出了托卡马克等离子体中漂移不稳定性的回旋动理学二维本征模积分方程组。该方程组保留了离子的动理学效应,包括沿磁场的运动、磁场梯度和曲率漂移以及有限拉莫半径效应。与传统的采用气球模表象得到的一维回旋动理学方程(其只能给出不稳定模沿磁场线的结构)不同,该方程组不仅能给出托卡马克等离子体中漂移不稳定模的径向结构,同时还考虑了由离子的环形性漂移引起的相邻极向模之间的线性耦合,进而得到模的极向结构。该结果为相应的数值模拟研究提供了理论基础。