Showing papers on "Spherical shell published in 1993"

Effects of an endothermic phase transition at 670 km depth in a spherical model of convection in the Earth's mantle

Abstract: Numerical modelling of mantle convection in a spherical shell with an endothermic phase change at 670 km depth reveals an inherently three-dimensional flow pattern, containing cylindrical plumes and linear sheets which behave differently in their ability to penetrate the phase change. The dynamics are dominated by accumulation of downwelling cold material above 670 km depth, resulting in frequent avalanches of upper-mantle material into the lower mantle. This process generates long-wavelength lateral heterogeneity, helping to resolve the contradiction between seismic tomographic observations and expectations from mantle convection simulations.

Visible photoluminescence from oxidized Si nanometer-sized spheres: Exciton confinement on a spherical shell.

Abstract: We report strong visible photoluminescence (PL) at room temperature from oxidized Si nanometer-sized spheres with a spherical crystalline Si (c-Si) core and an amorphous ${\mathrm{SiO}}_{2}$ (a-${\mathrm{SiO}}_{2}$) surface layer. The peak energy of the broad PL spectrum is about 1.65 eV, which is independent of the core diameter. We propose a model in which excitons are confined on a spherical shell, an interfacial layer between the c-Si core and the a-${\mathrm{SiO}}_{2}$ surface layer, and in which the exciton confinement enhances the oscillator strength and the PL intensity.

Acoustic radiation pressure acting on spherical and cylindrical shells

Abstract: The acoustic radiation pressure resulting from a plane wave incident on spherical shells and cylindrical shells immersed in a fluid is investigated theoretically in relation to the thickness of the shell and the contents of the hollow region. The results for the frequency dependence of acoustic radiation pressure computed on the basis of the theory are demonstrated for stainless shells with the hollow region filled water or air. Significant differences in acoustic radiation pressure occur when the interior region is changed from water to air.

Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number

Abstract: We examine thermal convection in a rotating spherical shell with a spatially non-uniformly heated outer surface, concentrating on three distinct heating modes: first, with wavelength and symmetry corresponding to the most unstable mode of the uniformly heated problem; secondly, with the critical wavelength but opposite equatorial symmetry; and thirdly, with wavelength much larger than that of the most unstable mode. Analysis is focused on boundary-locked convection, the associated spatial resonance phenomena, the stability properties of the resonance solution, and time-dependent secondary convection. A number of new forms of instability and convection are found: the most interesting is perhaps the saddle-node bifurcation, which is the first to be found for realistic fluid systems governed by partial differential equations. An analogous Landau amplitude equation is also analysed, providing an important mathematical framework for understanding the complicated numerical solutions.

Ultrasonic scattering from spherical shells including viscous and thermal effects

Abstract: An analysis for sound scattering and attenuation by shell structures immersed in fluids is outlined. While the form of the analysis permits easy comparison with previous results for simple spheres, it includes novel features of viscoelasticity in the shell and core materials (which may be fluid or solid), viscosity in the suspending fluid, and thermal effects. The computational procedure adopted is outlined with particular reference to the avoidance of numerical instabilities at very low and very high values of ‘‘ka.’’ The testing of the program by comparison with previous results on simpler systems is reported, and a limited selection of numerical results is given. The program has been used successfully over the range of ka values between 10−6 and 200, its robustness needs closer definition.

Excitation of a fluid‐filled, submerged spherical shell by a transient acoustic wave

Abstract: A mathematical formulation and method of solution for the title problem are developed, and numerical results are presented for excitation by a plane step wave. The formulation is based on the familiar equations of motion for a thin spherical shell and the wave equation for the internal and external fluid domains. The Laplace transform is invoked, and the usual separation of variables method produces modal equations for each component of the Fourier–Legendre series solution. The modal equations are then restructured to facilitate transform inversion, yielding delayed differential equations in time for each response mode of the shell‐fluid system, which are integrated numerically in time. Complete response solutions then follow by modal superposition, with special techniques being employed to improve modal convergence. Transient response histories are shown for a step‐wave‐excited steel shell containing water and submerged in water. Also, these results are compared with their counterparts for an empty submerged steel shell.

Gravitational Collapse in an Expanding Universe: Asymptotic Self-similar Solutions

Abstract: The collapse and virialization of self-similar spherical density perturbations to a flat Einstein-de Sitter universe is investigated and in particular the asymptotic similarity solutions. Previous calculations of Gunn and Fillmore & Goldreich of the asymptotic solutions are improved here and the collapse factor of a given spherical shell is calculated. Within the framework of the asymptotic behavior exact analytical solutions are obtained

Transitions to chaotic thermal convection in a rapidly rotating spherical fluid shell

Abstract: Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell heated from below and within have been carried out with a nonlinear, three-dimensional, time-dependent pseudospectral code. The investigated phenomena include the sequence of transitions to chaos and the differential mean zonal rotation. At the fixed Taylor number T a =106 and Prandtl number Pr=1 and with increasing Rayleigh number R, convection undergoes a series of bifurcations from onset of steadily propagating motions SP at R=R c = 13050, to a periodic state P, and thence to a quasi-periodic state QP and a non-periodic or chaotic state NP. Examples of SP, P, QP, and NP solutions are obtained at R = 1.3R c , R = 1.7 R c , R = 2R c , and R = 5 R c , respectively. In the SP state, convection rolls propagate at a constant longitudinal phase velocity that is slower than that obtained from the linear calculation at the onset of instability. The P state, characterized by a single frequency and its harmonics, has ...

Highly supercritical thermal convection in a rotating spherical shell: Centrifugal vs. radial gravity

Abstract: Numerical calculations of fully-developed thermal convection in a rapidly rotating spherical shell are presented. We analyze the three-dimensional structure of convection driven by basal heating at Rayleigh number Ra = 6.67 × 106 and Ekman number E=10−4 in a shell with the geometry of the Earth's liquid outer core. Calculations using two different body force distributions are compared: radial gravity and centrifugal acceleration. In both cases the flow consists of a chaotic array of small-scale, time-dependent columnar vortices aligned parallel to the rotation axis plus large-scale zonal and meridional circulations. The columnar vortices are driven by thermal plumes localized in the equatorial region. The zonal flow consists mainly of two parts: a cylindrical (geostrophic) part driven by Reynolds stresses derived from the small-scale convection and a thermal wind driven by large-scale temperature gradients. For both body force distributions, convective heat transfer occurs mostly in the equatoria...

Creeping Flow Past a Porous Spherical Shell

Abstract: An exact solution for the creeping flow past a porous spherical shell has been obtained using the Stokes & Brinkman equations. Expressions for the velocities and pressure fields both inside and outside of porous spherical shell are obtained and the force acting on the shell is also determined. It is found that the drag on a porous spherical shell is smaller as compared to the shell with solid core. Several limiting cases of the solution are also discussed. In particular it is shown that the solution obtained using Darcy's law can be derived easily with the appropriate choice of boundary conditions. Eine exakte Losung der schleichenden Stromung hinter einer porosen Kugelschale wird unter Benutzung der Stokes- und Brinkmangleichungen abgeleitet. Ausdrucke fur die Geschwindigkeiten und das Druckfeld innerhalb und auserhalb der porosen Kugelschale werden erhalten, und die auf die Schale wirkende Kraft wird ebenfalls bestimmt. Es zeigt sich, das der Stromungswiderstand auf eine porose Kugelschale, verglichen mit einer Schale mit festem Kern, kleiner ist. Verschiedene Grenzfalle der Losung werden diskutiert. Insbesondere wird gezeigt, das die Losung, die aus dem Darcyschen Gesetz abgeleitet wird, leicht durch passende Wahl der Randbedingungen erhalten werden kann.

Three‐dimensional spherical models of layered and whole mantle convection

Abstract: We present numerical calculations of three-dimensional spherical shell thermal convection for constant viscosity and stratified viscosity models of whole-layer and two-layer mantle convection. These four examples are intended to provide theoretical guidance for determining the style of convection that is occurring in Earth's mantle. An impermeable interface between the upper and lower convecting shells in the two-layer solutions is placed at a depth of 670 km to coincide with the mantle seismic discontinuity that divides the upper and lower mantle. The interface results in an internal thermal boundary layer that raises the mean temperature in the lower shell by about 1400 K compared to the whole-layer solutions. The patterns of convection in the upper part of the whole-layer solutions are dominated by narrow arcuate sheetlike downflows in a background of weak upflow. In contrast, the upper shells of the two-layer solutions have complicated networks of convective rolls with the upflows and downflows having very similar structure. The structure of convection in the lower shells is similar to that in the lower part of the whole-layer solutions. Based on the horizontal structure of subduction zones on Earth's surface and on tomographic images of temperature variations in Earth's mantle, we conclude that the style of convection in Earth's mantle is more like that of the whole-mantle models.

Simulation study of a magnetohydrodynamic dynamo : convection in a rotating spherical shell

Abstract: Numerical simulations on the thermal convection of a neutral fluid (without a magnetic field) in a rotating spherical shell have been carried out. The results indicate that sufficiently rapid rotation results in strong differential rotation, with a pronounced equatorial acceleration. The formation dynamics of convection columns aligned to the rotation axis is studied extensively. A new generation mechanism of differential rotation is then proposed which concludes that the fluid motion generates an equatorial acceleration by selectively exciting the cyclonic columns in the spherical shell.

Observation of the midfrequency enhancement of tone bursts backscattered by a thin spherical shell in water near the coincidence frequency

Abstract: A prominent feature predicted for the backscattering of tone bursts by thin spherical shells is an enhancement of a guided wave contribution in the midfrequency range [L. G. Zhang et al., J. Acoust. Soc. Am. 91, 1862–1874 (1992)]. This feature may be useful for certain inverse problems and is associated with a strongly coupled slightly subsonic wave denoted by some authors as the a0− wave. The present research gives a comparison between a ray approximation and experiments in which tone bursts having carrier frequencies in the range 35

Steady, three-dimensional, internally heated convection

Abstract: Numerical calculations have been carried out of steady, symmetric, three‐dimensional modes of convection in internally heated, infinite Prandtl number, Boussinesq fluids at a Rayleigh number of 1.4×104 in a spherical shell with inner/outer radius of 0.55 and in a 3×3×1 rectangular box. Multiple patterns of convection occur in both geometries. In the Cartesian geometry the patterns are dominated by cylindrical cold downflows and a broad hot upwelling. In the spherical geometry the patterns consist of cylindrical cold downwellings centered either at the vertices of a tetrahedron or the centers of the faces of a cube. The cold downflow cylinders are immersed in a background of upwelling within which there are cylindrical hot concentrations (plumes) and hot halos around the downflows. The forced hot upflow return plumes of internally heated spherical convection are fundamentally different from the buoyancy‐driven plumes of heated from below convection.

Changing the Inertial Mass of a Charged Particle

Abstract: We calculate utilizing Weber's law the force on a moving charge exerted by a stationary charged spherical shell surrounding it. We obtain a net force different from zero which is proportional to the acceleration of the test particle relative to the spherical shell. This result can be interpreted by saying that the inertial mass of a test particle should change if it is placed inside a charged spherical shell. We conclude that this modification in the inertial mass is proportional to the electrostatic potential of the charged spherical shell and to the electric charge of the test particle. Then we present some possible experiments which could be performed to test this prediction.

Variational formulation of acoustic radiation from submerged spheroidal shells

Abstract: The surface variational principle (SVP) governing acoustic interaction between a vibrating surface and a surrounding fluid is combined with the dynamic equations governing response of a shell of revolution subjected to axisymmetric harmonic excitation. Ritz series representations are used to represent the spatial dependence of the surface pressure and shell displacement components. Two formulations are presented, with the difference being whether an intermediate computation of the in‐vacuo modes is performed. The direct approach, in which the Ritz coefficients are determined directly from the coupled formulation, is used to validate the SVP approach for the case of a spherical shell, whose response is known analytically. For the case of a slender spheroidal shell, the direct approach is compared to the results obtained from the modal approach, in which a truncated set of in‐vacuo modes forms the basis functions representing displacement. The aspect ratio and shell properties for this evaluation are select...

Numerically stable global matrix approach to radiation and scattering from spherically stratified shells

Abstract: Although the theoretical formulation in terms of spherical harmonics for seismoacoustic propagation in spherically stratified, fluid/elastic media has been known for many decades, its numerical implementation is nontrivial. Thus, the limited machine precision is incapable of directly representing the dynamic range of the spherical harmonics throughout the stratification. The numerical solution for the expansion coefficients is therefore inherently unstable. However, this problem is equivalent to the numerical stability issue facing numerical implementation of the depth‐separated wave equation for plane‐stratified media. There, numerical stability has been obtained by properly factorizing the depth exponentials, and a number of stable algorithms are available. Here, an equivalent factorization of the spherical harmonics is described. The formulation is directly compatible with the global matrix approach used in the SAFARI plane‐stratified model, and this code has therefore been straightforwardly modified to provide unconditionally stable solutions for arbitrary fluid/elastic, spherical stratifications. The performance of the model is illustrated by a few numerical examples of relevance to structural acoustics.

Radiative-transfer in a linearly-anisotropic spherical medium

Abstract: The radiative-transfer problem in an absorbing, emitting, anisotropically scattering, inhomogenous, spherical shell is considered. The medium is taken to have diffusely reflecting boundaries and an internal energy source. The Galerkin method is used to calculate the partial heat fluxes at the boundaries, the radiation density and the net radiant heat flux through the medium. The calculations are carried out for isotropic scattering, forward and backward anisotropic scattering, for linear and quadratic space-dependent, single-scattering albedos, and for different internal sources.

Nonlinear planetary dynamos in a rotating spherical shell. III. α2ω models and the geodynamo

Abstract: The observed westward drift of some patches of the Earth's magnetic field suggests that the geodynamo is of the αω kind. However solutions of the αω dynamo equations tend to oscillate on a very rapid time scale, in disagreement with the observed long times between reversals. It is suggested that the problem can be cured by taking into account the generation of toroidal field by the α effect. The resulting α2ω dynamo equations are solved in the geometry of a spherical shell in both the linear and the nonlinear regime. Steady and oscillatory Taylor state solutions are found. The slow time dependence of the oscillatory solutions compares well with the observed frequency of geomagnetic reversals.

Local temporal variance of Wigner’s distribution function as a spectroscopic observable: Lamb wave resonances of a spherical shell

Abstract: The time‐frequency analysis of the transient response of elastic objects has been asserted to be useful for relating features of the scattering to physical parameters of the scatterer. [N. Yen, L. R. Dragonette, and S. K. Numrich, J. Acoust. Soc. Am. 87, 2359–2370 (1990)]. In this article, the Wigner distribution function is applied to the mean‐square temporal spread (as a function of frequency) of the impulse response of a scatterer. The application is illustrated for a thin stainless steel shell in water. The emphasis is on resonance frequencies, where the temporal spread can be large, and locally the frequency response is taken to have a Breit–Wigner form. That approximation yields a simple relationship between the measure of the temporal spread, Q(ka), and the width of the resonance when evaluated at the resonance frequency. The frequency dependence is expressed in terms of the wave number–radius product ka for the shell. The analysis is supported by calculation of resonance widths from ray theory and by ray approximations of form function contributions that dominate Q(ka) near certain resonances.

Linear and weakly nonlinear variable viscosity convection in spherical shells

Abstract: A theoretical study of linear and weakly nonlinear thermal convection in a spherical shell is performed. The Boussinesq fluid is of infinite Prandtl number and its viscosity is temperature dependent. The linear stability eigenvalue problem is derived and solved by a shooting method assuming isothermal, stress-free boundaries, a self-gravitating fluid, and corresponding to two heating models. The first is heating from below, and the second is a model of combined heating from below and within, such that convection is described by a self-adjoint linear stability formulation. In addition, nonlinear, hemispherical, axisymmetric convection is computed by a finite volume technique for a shell with 0.5 aspect ratio. It is shown that 2-cell convection occurs as transcritical bifurcation for a viscosity constrast across the shell up to about 150. Motions with four cells are also possible. As expected, the subcritical range is found to increase with increasing viscosity contrast, even when the linear operator is self-adjoint.

Sealing cover for coupling

Abstract: A sealing cover fitted on a coupling which connects an input shaft (1) and an output shaft (2) to each other for relative movement as in a U-joint including an outer member (3) having an outer spherical surface (3a) and connected to one of said shafts (2), comprises a first spherical shell (4) having an outer part made partly of a low flexibility material and having an inside part spherical surface slidably fitted on the outer spherical surface (3a), and a part spherical channel (7) open at the end of the shell near the other of the shafts (1), a second part spherical shell (8) slidably inserted at one end thereof into the channel (7) and fitted at the other end thereof on the shaft (1), a first sealing portion provided near one end of the first shell (4) in contact with the outer surface (3a) of the outer member (3), and a second sealing portion (10) provided near the other end of the first shell in contact with the outer part spherical surface (8c) of the second shell (8).