The general solution of the HENON–HEILES problem     

K.?E.?PapadakisEmail author  C.?L.?Goudas  G.?A.?Katsiaris 《Astrophysics and Space Science》2005,295(3):375-396

The general solution of the Henon–Heiles system is approximated inside a domain of the (x, C) of initial conditions (C is the energy constant). The method applied is that described by Poincaré as ‘the only “crack” permitting penetration into the non-integrable problems’ and involves calculation of a dense set of families of periodic solutions that covers the solution space of the problem. In the case of the Henon–Heiles potential we calculated the families of periodic solutions that re-enter after 1–108 oscillations. The density of the set of such families is defined by a pre-assigned parameter ε (Poincaré parameter), which ascertains that at least one periodic solution is computed and available within a distance ε from any point of the domain (x, C) for which the approximate general solution computed. The approximate general solution presented here corresponds to ε = 0.07. The same solution is further improved by “zooming” into four square sub-domain of (x, C), i.e. by computing sufficient number of families that reduce the density parameter to ε = 0.003. Further zooming to reduce the density parameter, say to ε = 10⁻⁶, or even smaller, although easily performable in both areas occupied by stable as well as unstable solutions, was found unnecessary. The stability of all members of each and all families computed was calculated and presented in this paper for both the large solution domain and for the sub-domains. The correspondence between areas of the approximate general solution occupied by stable periodic solutions and Poincaré sections with well-aligned section points and also correspondence between areas occupied by unstable solutions and Poincaré sections with randomly scattered section points is shown by calculating such sections. All calculations were performed using the Runge-Kutta (R-K) 8th order direct integration method and the large output received, consisting of many thousands of families is saved as “Atlas of the General Solution of the Henon–Heiles Problem,” including their stability and is available at request. It is concluded that approximation of the general solution of this system is straightforward and that the chaotic character of its Poincaré sections imposes no limitations or difficulties.  相似文献   

Flux–Profile Relationships Over a Fetch Limited Beech Forest     

Ebba Dellwik  Niels Otto Jensen 《Boundary-Layer Meteorology》2005,115(2):179-204

The influence of an internal boundary layer and a roughness sublayer on flux–profile relationships for momentum and sensible heat have been investigated for a closed beech forest canopy with limited fetch conditions. The influence was quantified by derivation of local scaling functions for sensible heat flux and momentum (_h and _m) and analysed as a function of atmospheric stability and fetch. For heat, the influences of the roughness sublayer and the internal boundary layer were in agreement with previous studies. For momentum, the strong vertical gradient of the flow just above the canopy top for some wind sectors led to an increase in _m, a feature that has not previously been observed. For a fetch of 500 m over the beech forest during neutral atmospheric conditions, there is no height range at the site where profiles can be expected to be logarithmic with respect to the local surface. The different influence of the roughness sublayer on _h and _m is reflected in the aerodynamic resistance for the site. The aerodynamic resistance for sensible heat is considerably smaller than the corresponding value for momentum.  相似文献   

Ionic association in H₂O–CO₂ fluids at mid-crustal conditions     

J. V. WALTHER 《Journal of Metamorphic Geology》1992,10(6):789-797

The equilibrium constant, K _a, of the association reaction to form ion pairs from charged solute species in supercritical solutions can be calculated from a model based on published equations. Log K _a at constant pressure is a linear function of the inverse in the dielectric constant of the fluid times temperature. The dielectric properties of H₂O and CO₂ at supercritical pressures and temperatures can also be evaluated using the Kirkwood equation. Using Looyenga mixing rules, the dielectric constant of H₂O–CO₂ mixtures can be obtained and the change in log K _a with addition of CO₂ in aqueous solutions evaluated. These changes in log K _a with addition of CO₂ are consistent with measured changes of log K _a with addition of Ar in supercritical H₂O–Ar solutions.
Log K _a of KCl and NaCl increase to an increasing extent as the mole fraction of CO₂ increases in H₂O–CO₂ solutions. For instance, at 2 kbar and constant temperature between 400 and 600° C, log K _a of KCl increases by about two orders of magnitude whilst that of NaCl increases by over four orders of magnitude as the CO₂ mole fraction increases from 0.0 to 0.35. Such changes in log K _a will have dramatic effects on the solubility of minerals in CO₂-rich environments.  相似文献   

Identification and evolution of hydrogeochemical processes in the groundwater environment in an area of the Palar and Cheyyar River Basins,Southern India   总被引：5，自引：0，他引：5  

N.?Rajmohan  L.?ElangoEmail author 《Environmental Geology》2004,46(1):47-61

The Palar and Cheyyar River Basins in Tamil Nadu state of Southern India are characterised by different geological formations, and groundwater is the major source for domestic, agricultural and other water-related activities. Hydrogeochemical studies were carried out in this area with the objective of identifying the geochemical processes and their relation to groundwater quality. Groundwater samples were collected once a month from 43 groundwater wells in this area from January 1998 to July 1999. Sampling procedures and chemical analysis were carried out as per the standard methods. Chemical data are used for mathematical calculations and graphical plots to understand the chemical process and its relation to the groundwater quality. The chemical composition of groundwater in the central part of the study area mainly depends on the recharge from lakes and the river, which is explained by a mixing mechanism. In addition, weathering of silicate minerals controls the concentration of major ions such as sodium, calcium, magnesium and potassium in the groundwater of this area. Further, the activity ratios indicate that the groundwater is in equilibrium with kaolinite, smectite and montmorrillonite. The reverse ion exchange process controls the concentration of calcium, magnesium and sodium in hard rock formations, and dissolution of carbonate minerals and accessory minerals is the source of Ca and Mg, in addition to cation exchange in the sedimentary formations. In general, the chemical composition of the groundwater in this area is influenced by rock–water interaction, dissolution and deposition of carbonate and silicate minerals, ion exchange, and surface water interactions.  相似文献   

Melt-Solid Dihedral Angles of Common Minerals in Natural Rocks   总被引：3，自引：0，他引：3  

HOLNESS  M. B. 《Journal of Petrology》2006,47(4):791-800

The melt–solid dihedral angle has been measured in a rangeof igneous rock types, ranging in composition from picrite,through basalt, phonolite, andesite and rhyolite, for the mineralsquartz, leucite, plagioclase, olivine, amphibole and clinopyroxene.Populations of up to 104 true 3-D angles were measured in eachsample using a universal stage mounted on an optical microscope.The median and standard deviation of the angle populations foreach mineral are distinct (plagioclase 25°, with standarddeviation (SD) 11°; clinopyroxene 38°, with SD 14°;olivine 29°, with SD 13°; quartz 18°, with SD 9°;leucite 20°, with SD 11°), with no control by eithermelt composition or degree of approach of the grains to theirequilibrium shapes. KEY WORDS: dihedral angle; textural equilibrium; universal stage  相似文献   

SPATIAL AND THERMAL CHARACTERISTICS OF MOUNTAIN PERMAFROST,NORTHWEST CANADA     

ANTONI G. LEWKOWICZ  PHILIP P. BONNAVENTURE  SHARON L. SMITH  ZOÉ KUNTZ 《Geografiska Annaler: Series A, Physical Geography》2012,94(2):195-213

  相似文献   

Turbulent Kinetic Energy Dissipation in the Surface Layer     

D. Charuchittipan  J. D. Wilson 《Boundary-Layer Meteorology》2009,132(2):193-204

We estimated the turbulent kinetic energy (TKE) dissipation rate for thirty-two 1-h intervals of unstable stratification covering the stability range 0.12 ≤ −z/L ≤ 43 (z/L is the ratio of instrument height to the Obukhov length), by fitting Kolmogorov’s inertial subrange spectrum to streamwise spectra observed over a desert flat. Estimated values are compatible with the existence of local equilibrium, in that the TKE dissipation rate approximately equalled the sum of shear and buoyant production rates. Only in the neutral limit was the turbulent transport term in the TKE budget measured to be small.  相似文献   

Artificial halo orbits for low-thrust propulsion spacecraft     

Shahid Baig  Colin R. McInnes 《Celestial Mechanics and Dynamical Astronomy》2009,104(4):321-335

We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L ₁ and L ₂ in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points inside L ₁ and beyond L ₂ where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion systems are obtained by using the Lindstedt Poincaré and differential corrector method respectively. Both the period and minimum amplitude of halo orbits about artificial equilibrium points inside L ₁ decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L ₂ in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first increases and then decreases after the thrust acceleration exceeds 0.415 mm/s². Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times and require a reasonably small low-thrust acceleration 0.0593 mm/s².  相似文献   

Interplay between equilibrium and kinetics in prograde metamorphism of pelites: an example from the Nelson aureole, British Columbia   总被引：2，自引：0，他引：2  

D. R. M. PATTISON  D. K. TINKHAM 《Journal of Metamorphic Geology》2009,27(4):249-279

The distribution of metapelitic mineral assemblages in the Nelson aureole, British Columbia, generally conforms to what is predicted from phase equilibria. However, in detail, the sequence and spacing of isograds, mineral textures and mineral compositions and mineral chemical zoning do not. Two of the main disequilibrium features in the aureole are: (i) delay in the onset and progress of several reactions, i.e. overstepping in temperature; and (ii) unreactivity of staurolite and especially garnet porphyroblasts when they are reactants in prograde reactions. The thermal overstepping is ascribed to difficulty of nucleation of the product porphyroblasts and sluggishness of dissolution of porphyroblasts when they are reactants. The extent to which these kinetic barriers delay the onset of reaction is related to the reaction affinity of each reaction, defined herein as the Gibbs free‐energy difference between the thermodynamically stable, but not‐yet‐crystallized, products and the metastable reactants. For oversteps in temperature (ΔT), reaction affinity is, in turn, related to the difference in entropy (ΔS) between these two states through the relation A = ΔT * ΔS. Mineral reactions which release large quantities of H₂O, such as chlorite‐consuming reactions, have a higher entropy change per unit of temperature overstep, and therefore a higher reaction affinity, than those which release little or no H₂O, such as the chlorite‐free staurolite‐consuming reaction. Thermal overstepping is consequently expected to be less for the former than for the latter, as was estimated in the aureole where 0 to 30 °C overstepping was required for garnet, staurolite and andalusite growth from a muscovite + chlorite‐bearing precursor rock and ～70 °C overstepping was required for the growth of Al₂SiO₅ from a staurolite‐bearing, chlorite‐free precursor. In all cases, reaction progress was strongly influenced by the presence or absence of fluid, with presence of fluid lowering kinetic barriers to nucleation and growth and therefore the degree of thermal overstepping. Textural features of rocks from the nearly coincident garnet, staurolite and andalusite isograds are suggestive of a fluid‐catalysed ‘cascade effect’ in which reaction took place in a narrow temperature interval; several competing muscovite + chlorite‐consuming reactions, some metastable, appear to have occurred in parallel. Metamorphic reaction, fluid release and possibly fluid presence in general in the aureole were episodic rather than continuous, and in several cases well removed from equilibrium conditions. The extent to which these findings apply to regional metamorphism depends on several factors, a major one being enhanced deformation, which is expected to lower kinetic barriers to nucleation and growth.  相似文献   

基于GeoStudio的边坡稳定性分析及支护方案选择的理论探讨     

何保  宋帅 《地质与勘探》2019,55(5):1329-1335

本文选择大连市中山区长利巷地区的边坡,通过现场采取岩土样,进行室内试验得出各地层力学物理参数,对边坡进行稳定性分析。针对目前所拥有的模拟软件与方法,分析并筛选出适合该边坡的模拟软件。利用GeoStudio软件建模、模拟,使用Morgenstern-Price方法进行计算,通过分析得到计算结果,该边坡的安全安全系数为1. 127,认为其属于不稳定边坡。对于不稳定边坡,本文对主流的边坡支护方法进行分析筛选,拟采用扶壁式挡土墙支护方案对其进行预防。根据规范设计挡土墙尺寸,并对该挡土墙进行抗倾覆抗滑移验算,得出的验算安全系数满足规范要求,由此可得知该支护方案方法合理,技术可行。  相似文献