旋转椭球面上的应变与转动张量表达

以旋转椭球体面上某点为原点建立一个大地坐标单位活动坐标架. 通过平移, 使活动坐标架的原点与以椭球中心为原点的笛卡尔单位标架的原点相重合. 然后再通过两次标架旋转, 使活动坐标架与笛卡尔单位标架完全重合. 本文给出了使两个单位标架相重合的转换关系式, 以及该点位移在两个单位标架中的坐标转换式; 在此基础上, 考虑该点的位移及活动坐标架皆为该点大地坐标的函数, 经复杂推导, 分别给出了该点位移向量的微分在大地坐标系中的分量以及该点分别沿坐标曲线的弧微分表达式, 继而导出了该点的位移梯度矩阵; 最后推导出了椭球坐标系的应变张量与转动张量表达式, 并对转动张量的几何含义进行了较详细的解释, 且采用曲面理论对球面与椭球面的应变张量间的内在关系进行了讨论.      

Qualitative and quantitative analysis of vorticity,strain and area change in general non-isochoric 2D deformation

A scale free representation of a general non-isochoric 2D deformation is presented which is amenable to mathematical analysis. By describing deformation in 2D in terms of polar coordinates the stretching and rotational histories of linear elements separate and are easily analysed both qualitatively and quantitatively. An analysis of finite strain combined with dynamical considerations allows the derivation of equations which may be used to estimate finite strain, area change and kinematic vorticity number. Numerical investigation of method developed here was carried out and it was found to perform well unless large area changes occur in combination with large components of simple shear. A re-analysis of natural data indicates the method is consistent.     

东中国海HYCOM模式垂向坐标对比研究

基于挪威南森环境遥感中心改进的NERSC-HYCOM 模式, 利用单向嵌套技术与欧洲中心提供的2008 年ERA-I 高分辨率强迫场针对东中国海及其邻近海区进行了不同垂向坐标配置的四个敏感性试验。通过分析东中国海区域的温度、盐度, 流速的分布和变化, 探讨了HYCOM 模式中不同垂向坐标设置对东中国海近岸区域的影响以及黑潮流速及路径对不同坐标设置的响应, 期望对HYCOM 模式更深入的研究提供参考。结果表明: (1)在东中国海区上层并不适于采用等密度坐标方案, 也就是说应该采用z坐标或σ坐标用以表征此处混合层的季节性变化特征; (2)针对东海大陆架区给出了10 个位置上的模式与浮标观测资料的温、盐平均误差(ME)、均方根差(RMS)及相关系数(R)指标, 发现对于不同区域, 每种试验的适用性都不同; (3)使用高频资料时, 模拟的流速普遍偏高, 东海黑潮冬夏路径的异同指出了σ-z-iso 与z-iso 试验模拟效果较好, 但模拟的日本岛南岸的弯曲流场位置偏南; 而z-only 试验模拟的日本岛以南的黑潮路径是有所改观的, z坐标的分辨率对表层的黑潮路径影响很大; σ-only 试验模拟的整个黑潮路径的效果最差。     

光速不变原理推广（Ⅱ）──论弯曲时空中光速不变

根据等效原理及理论自洽的要求，把光速不变原理推广到弯曲时空。并在此基础上阐明了坐标、坐标变换，物理量的描述与测量等最基本问题的物理意义。     

A projective approach to orbit determination from three sight-lines

Concepts from projective geometry are used to provide a coherent framework for the determination of orbits from observation data comprising lines of sight at three known times. A novel way of presenting the results in a finite diagram is introduced, The effectiveness of the approach is demonstrated by an example, using a simple spreadsheet. A computer-graphic implementation is recommended.     

空间管治视角下京津冀协同发展类型区划

京津冀协同发展的核心是基于问题导向和底线思维,打破行政区划,在更大区域尺度上优化资源配置,实现区域整体发展目标。因此,从优化空间规划体系的高度,开展京津冀区域协同发展的类型区划和类型区管理,实现分区施策的精细化管治极为重要。本文首先立足于京津冀区域发展差异,利用空间属性双聚类的方法将京津冀划分为中部核心功能引领区、东部沿海重点发展区、南部门户功能拓展区、西部和北部生态涵养保护区等四大区域;然后以区县为最小分析单元,从现状开发强度、用地增量预测和生态保护责任等三大维度构建类型划分指标体系,并利用三维空间坐标划分方法将京津冀划分为五大类型区,即：城镇优化发展区、城镇重点拓展区、现代农业发展区、适度建设发展区和严格生态保护区;最后,在空间管治视角下提出京津冀区域分区管治与区域协同管理的建议。     

Theory of anisotropic dynamic ray tracing in ray-centred coordinates

A 4×4-propagator matrix formalism is presented for anisotropic dynamic ray tracing, including the propagation across curved interfaces. The computations are organised in the same way as in ervený's well-known isotropic propagator matrix formalism. Attention is paid to cases where double eigenvalues of the Christoffel matrix result in unstable expressions in the dynamic ray tracing system, but where geometrical spreading is well-behaved.     

Inertial Instability of Arbitrarily Meandering Currents Governed by the Eccentrically Cyclogeostrophic Equation

Using natural coordinates, we have derived a criterion for the inertial instability of arbitrarily meandering currents. Such currents, governed by the eccentrically cyclogeostrophic equation, are adopted as the basic current field for the parcel method. We assume that any virtual displacement which is given to a water parcel moving in the basic field has no influence on this field. From the conservation of mechanical energy for a virtual displacement we derive an inertial instability frequency ω _m = [(f + 2u/r)Z]^0.5 for the eccentrically cyclogeostrophic current, where f is the Coriolis parameter, u the velocity (always positive), r the radius of curvature of a streamline (negative for an anticyclonic meander), and Z the vertical component of absolute vorticity. If ω _m ² is negative, the eccentrically cyclogeostrophic current becomes unstable. Although the conventional, centrifugal instability criterion, derived from the conservation of angular momentum in a circularly symmetric current field, has a certain meaning for a monopolar vortex, it contains a radial shear vorticity that is difficult to use in arbitrarily meandering currents. The new criterion ω _m ² contains a lateral shear vorticity that is applicable to arbitrarily meandering currents. Examining instabilities of concentric rings with radii of 50–100 km, we consider reasons why the anticyclonic supersolid rotation has been very much less frequently observed than the cyclonic supersolid rotation, despite a prediction of some common stability and a rapid change in radial velocity gradient for the former. Classifying eccentric streamlines into the large and small curvature-gradient types, we point out that the large-gradient curvature in anticyclonic rings is apt to be unstable. This revised version was published online in July 2006 with corrections to the Cover Date.     

A generalized coordinate ocean model and a comparison of the bottom boundary layer dynamics in terrain-following and in z-level grids

Sensitivity studies with a new generalized coordinate ocean model are performed in order to compare the behavior of bottom boundary layers (BBLs) when terrain-following (sigma or combined sigma and z-level) or z-level vertical grids are used, but most other numerical aspects remain unchanged. The model uses a second-order turbulence closure scheme that provides surface and BBL mixing and results in a quite realistic climatology and deep water masses after 100 year simulations with a coarse resolution (1° × 1°) basin-scale terrain-following grid. However, with the same turbulence scheme but using a z-level grid, the model was unable to produce dense water masses in the deep ocean. The latter is a known problem for coarse resolution z-level models, unless they include highly empirical BBL schemes.A set of dense water overflow experiments with high-resolution grids (10 and 2.5 km) are used to investigate the influence of model parameters such as horizontal diffusivity, vertical mixing, horizontal resolution, and vertical resolution on the simulation of bottom layers for the different coordinate systems. Increasing horizontal diffusivity causes a thinner BBL and a bottom plume that extends further downslope in a sigma grid, but causes a thicker BBL and limited downslope plume extension in a z-level grid. A major difference in the behavior of the BBL in the two grids is due to the larger vertical mixing generated by the turbulence scheme over the step-like topography in the z-level grid, compared to a smaller vertical mixing and a more stably stratified BBL in the sigma grid. Therefore, the dense plume is able to maintain its water mass better and penetrates farther downslope in the sigma grid than in the z-level grid. Increasing horizontal and vertical resolution in the z-level grid converges the results toward those obtained by a much coarser resolution sigma coordinate grid, but some differences remain due to the basic differences in the mixing process in the BBL.     

Dispersion and stability analyses of the linearized two-dimensional shallow water equations in Cartesian coordinates

In the present study, a Fourier analysis is used to develop expressions for phase and group speeds for both continuous and discretized, linearized two-dimensional shallow water equations, in Cartesian coordinates. The phase and group speeds of the discrete equations, discretized using a three-point scheme of second order, five-point scheme of fourth order and a three-point compact scheme of fourth order in an Arakawa C grid, are calculated and compared with the corresponding values obtained for the continuous system. The three-point second-order scheme is found to be non-dispersive with grid resolutions greater than 30 grids per wavelength, while both the fourth-order schemes are non-dispersive with grid resolutions greater than six grids per wavelength. A von Neumann stability analysis of the two- and three-time-level temporal schemes showed that both schemes are stable. A wave deformation analysis of the two-time-level Crank–Nicolson scheme for one-dimensional and two-dimensional systems of shallow water equations shows that the scheme is non- dispersive, independent of the Courant number and grid resolution used. The phase error or the dispersion of the scheme decreases with a decrease in the time step or an increase in grid resolution.