The Ideal Resonance Problem in its normal form is defined by the Hamiltonian(1) $$F = A (y) + 2B (y) sin^2 x$$ with(2) $$A = 0(1),B = 0(\varepsilon )$$ where ? is a small parameter, andx andy a pair of canonically conjugate variables. A solution to 0(?1/2) has been obtained by Garfinkel (1966) and Jupp (1969). An extension of the solution to 0(?) is now in progress in two papers ([Garfinkel and Williams] and [Hori and Garfinkel]), using the von Zeipel and the Hori-Lie perturbation methods, respectively. In the latter method, the unperturbed motion is that of a simple pendulum. The character of the motion depends on the value of theresonance parameter α, defined by(3) $$\alpha = - A\prime /|4A\prime \prime B\prime |^{1/2} $$ forx=0. We are concerned here withdeep resonance,(4) $$\alpha< \varepsilon ^{ - 1/4} ,$$ where the classical solution with a critical divisor is not admissible. The solution of the perturbed problem would provide a theoretical framework for an attack on a problem of resonance in celestial mechanics, if the latter is reducible to the Ideal form: The process of reduction involves the following steps: (1) the ration1/n2 of the natural frequencies of the motion generates a sequence.(5) $$n_1 /n_2 \sim \left\{ {Pi/qi} \right\},i = 1, 2 ...$$ of theconvergents of the correspondingcontinued fraction, (2) for a giveni, the class ofresonant terms is defined, and all non-resonant periodic terms are eliminated from the Hamiltonian by a canonical transformation, (3) thedominant resonant term and itscritical argument are calculated, (4) the number of degrees of freedom is reduced by unity by means of a canonical transformation that converts the critical argument into an angular variable of the new Hamiltonian, (5) the resonance parameter α (i) corresponding to the dominant term is then calculated, (6) a search for deep resonant terms is carried out by testing the condition (4) for the function α(i), (7) if there is only one deep resonant term, and if it strongly dominates the remaining periodic terms of the Hamiltonian, the problem is reducible to the Ideal form. 相似文献
The effect of the 11-year solar cycle on the response of planetary wavenumbers 1 and 2 at 10 and 30 hPa in winter to solar activity oscillations on the time scale of the Sun's rotation (27.2 day) is discussed in terms of statistical spectral analysis. The three oscillations studied are the 27.2 d (period of the Sun's rotation), 25.3 d (periodicity caused by modulation of the 27.2 d stratospheric response by annual atmospheric variation), and 54.4 d (doubled period of the solar rotation). A significant effect of the 11-year solar cycle is found for the 54.4 d periodicity in planetary wavenumber 1, and for the 27.2 and 25.3 d periodicities in planetary wavenumber 2. The effect of the 11-year solar cycle is expressed in the evident differences between the amplitudes of responses of planetary waves at maximum and minimum of the solar cycle: the amplitudes are much larger at high than at low solar activity. The 11-year modulation of planetary wave activity is most pronounced at mid-latitudes, mainly at 40–60°N, where the observed variability of planetary waves is large. The results obtained are in good agreement with results of the recent modeling study by Shindell et al. (Science 284 (1999) 305). 相似文献
In a previous publication (1977) the author has constructed a family () of long-periodic orbits in the Trojan case of the restricted problems of three bodies. Here he constructs the domain of the analytical solution of the problem of the motion, excluding the vicinity of thecritical divisor which vanishes at the exact commensurability of the natural frequencies 1 and 2. In terms of thecritical masses mj(2), or the associatedcritical energiesj2
(m), is the intersection of the intervals ofshallow resonance, of the form. Inasmuch as the intervals |2–
j2
|<j ofdeep resonance aredisjoint, it follows that (1) the disjointed family () embraces the tadpole branch, 021, lying in: and (2) despite the clustering of
j2
(m) atj=, the family () includes, for 2=1, an asymptoticseparatrix that terminates the branch in the vicinity of the Lagrangian pointL3.In a similar manner, the family () can be extended to the horseshoe branch 1<222
. 相似文献
On- and off-mound sediment cores from Propeller Mound (Hovland Mound province, Porcupine Seabight) were analysed to understand
better the evolution of a carbonate mound. The evaluation of benthic foraminiferal assemblages from the off-mound position
helps to determine the changes of the environmental controls on Propeller Mound in glacial and interglacial times. Two different
assemblages describe the Holocene and Marine Isotope Stage (MIS) 2 and late MIS 3 (∼31 kyr BP). The different assemblages
are related to changes in oceanographic conditions, surface productivity and the waxing and waning of the British Irish Ice
Sheet (BIIS) during the last glacial stages. The interglacial assemblage is related to a higher supply of organic material
and stronger current intensities in water depth of recent coral growth. During the last glaciation the benthic faunas showed
high abundances of cassidulinid species, implying cold bottom waters and a reduced availability of organic matter. High sedimentation
rates and the domination of Elphidium excavatum point to shelf erosion related to sea-level lowering (∼50 m) and the progradation of the BIIS onto the shelf. A different
assemblage described for the on-mound core is dominated by Discanomalina coronata, Gavelinopsis translucens, Planulina ariminensis, Cibicides lobatulus and to a lower degree by Hyrrokkin sarcophaga. These species are only found or show significantly higher relative abundances in on-mound samples and their maximum contribution
in the lower part of the record indicates a higher coral growth density on Propeller Mound in an earlier period. They are
less abundant during the Holocene, however. This dataset portrays the boundary conditions of the habitable range for the cold-water
coral Lophelia pertusa, which dominates the deep-water reefal ecosystem on the upper flanks of Propeller Mound. The growth of this ecosystem occurs
during interglacial and interstadial periods, whereas a retreat of corals is documented in the absence of glacial sediments
on-mound. Glacial conditions with cold intermediate waters, a weak current regime and high sedimentation rates provide an
unfavourable environmental setting for Lophelia corals to grow. A Late Pleistocene decrease is observed in the mound growth for Propeller Mound, which might face its complete
burial in the future, as it already happened to the buried mounds of the Magellan Mound province further north. 相似文献
A second-order exact expression for the evolution of probability density function of stress is derived for general, one-dimensional
(1-D) elastic–plastic constitutive rate equations with uncertain material parameters. The Eulerian–Lagrangian (EL) form of
Fokker–Planck–Kolmogorov (FPK) equation is used for this purpose. It is also shown that by using EL form of FPK, the so called
“closure problem” associated with regular perturbation methods used so far, is resolved too. The use of EL form of FPK also
replaces repetitive and computationally expensive deterministic elastic–plastic computations associated with Monte Carlo technique.
The derived general expressions are specialized to the particular cases of point location scale linear elastic and elastic–plastic
constitutive equations, related to associated Drucker–Prager with linear hardening. In a companion paper, the solution of
FPK equations for 1D is presented, discussed and illustrated through a number of examples. 相似文献
Elastic properties of fluid saturated porous media with aligned fractures can be studied using the model of fractures as linear-slip interfaces in an isotropic porous background. Such a medium represents a particular case of a transversely isotropic (TI) porous medium, and as such can be analyzed with equations of anisotropic poroelasticity. This analysis allows the derivation of explicit analytical expressions for the low-frequency elastic constants and anisotropy parameters of the fractured porous medium saturated with a given fluid. The five elastic constants of the resultant TI medium are derived as a function of the properties of the dry (isotropic) background porous matrix, fracture properties (normal and shear excess compliances), and fluid bulk modulus. For the particular case of penny-shaped cracks, the expression for anisotropy parameter ε has the form similar to that of Thomsen [Geophys. Prospect. 43 (1995) 805]. However, contrary to the existing view, the compliance matrix of a fluid-saturated porous-fractured medium is not equivalent to the compliance matrix of any equivalent solid medium with a single set of parallel fractures. This unexpected result is caused by the wave-induced flow of fluids between pores and fractures. 相似文献
We design a velocity–porosity model for sand-shale environments with the emphasis on its application to petrophysical interpretation of compressional and shear velocities. In order to achieve this objective, we extend the velocity–porosity model proposed by Krief et al., to account for the effect of clay content in sandstones, using the published laboratory experiments on rocks and well log data in a wide range of porosities and clay contents. The model of Krief et al. works well for clean compacted rocks. It assumes that compressional and shear velocities in a porous fluid-saturated rock obey Gassmann formulae with the Biot compliance coefficient. In order to use this model for clay-rich rocks, we assume that the bulk and shear moduli of the grain material, and the dependence of the compliance on porosity, are functions of the clay content. Statistical analysis of published laboratory data shows that the moduli of the matrix grain material are best defined by low Hashin–Shtrikman bounds. The parameters of the model include the bulk and shear moduli of the sand and clay mineral components as well as coefficients which define the dependence of the bulk and shear compliance on porosity and clay content. The constants of the model are determined by a multivariate non-linear regression fit for P- and S-velocities as functions of porosity and clay content using the data acquired in the area of interest. In order to demonstrate the potential application of the proposed model to petrophysical interpretation, we design an inversion procedure, which allows us to estimate porosity, saturation and/or clay content from compressional and shear velocities. Testing of the model on laboratory data and a set of well logs from Carnarvon Basin, Australia, shows good agreement between predictions and measurements. This simple velocity-porosity-clay semi-empirical model could be used for more reliable petrophysical interpretation of compressional and shear velocities obtained from well logs or surface seismic data. 相似文献
The Oshurkovo Complex is a plutonic sheeted complex which represents numerous successive magmatic injections into an expanding system of subparallel and subvertical fractures. It comprises a wide range of rock types including alkali monzodiorite, monzonite, plagioclase-bearing and alkali-feldspar syenites, in the proportion of about 70% mafic rocks to 30% syenite. We suggest that the variation within the complex originated mainly by fractional crystallization of a tephrite magma.
The mafic rocks are considered as plutonic equivalents of lamprophyres. They exhibit a high abundance of ternary feldspar and apatite, the latter may attain 7–8 vol.% in monzodiorite. Ternary feldspar is also abundant in the syenites. The entire rock series is characterized by high Ba and Sr concentrations in the bulk rock samples (3000–7000 ppm) and in feldspars (up to 1 wt.%). The mafic magma had amphibole at the liquidus at 1010–1030 °C based on amphibole geothermometer. Temperatures as low as this were due to high H2O and P2O5 contents in the melt (up to 4–6 and 2 wt.%, respectively). Crystallization of the syenitic magmas began at about 850 °C (based on ternary feldspar thermometry). The series was formed at an oxygen fugacity from the NNO to HM buffer, or even higher.
The evolution of the alkali monzodiorite–syenite series by fractional crystallization of a tephritic magma is established on the basis of geological, mineralogical, geochemical and Sm–Nd and Rb–Sr isotope data. The geochemical modeling suggests that fractionation of amphibole with subordinate apatite from the tephrite magma leaves about 73 wt.% of the residual monzonite melt. Further extraction of amphibole and plagioclase with minor apatite and Fe–Ti oxides could bring to formation of a syenite residuum. Rb–Sr isotopic analyses of biotite, apatite and whole-rock samples constrain the minimum age of basic intrusions at ca. 130 Ma and that of cross-cutting granite pegmatites at ca. 120 Ma. Hence the entire evolution took place in an interval of ≤10 My. Initial 87Sr/86Sr ratios for the mafic rocks range from 0.70511 to 0.70514, and for syenites from 0.70525 to 0.70542. Initial Nd (130 Ma) values for mafic rocks vary from −1.9 to −2.4, and for syenites from −2.9 to −3.5. In a Nd(T) vs. (87Sr/86Sr)i diagram, all rock types of the complex fall in the enriched portion of the Mantle Array, suggesting their derivation from a metasomatized mantle source. However, the small but distinguishable difference in Sr and Nd isotopic compositions between mafic rocks and syenites probably resulted from mild (10–20%) crustal contamination during differentiation. Large negative Nb anomalies are interpreted as a characteristic feature of the source region produced by Precambrian fluid metasomatism above a subduction zone rather than by crustal contamination. 相似文献