Showing papers on "Hydrostatic equilibrium published in 1989"

Hydrostatic airflow over mountains

Abstract: Publisher Summary This chapter describes the hydrostatic airflow over mountains. The flow of a density stratified fluid over an obstacle has been widely studied because of its application to atmospheric airflow over the irregular terrain of the earth's surface. Theories of vertically propagating waves, wave breaking, flow splitting, and wake eddies are discussed. The knowledge of these phenomena is used to construct an approximate regime diagram depicting how the nature of mountain-induced flow disturbance depends on the control parameters of the problem. The starting point for the discussion is the set of governing equations for a stratified Boussinesq fluid. It is found that on the top, lateral, and downstream boundaries, radiation or free-advection conditions might have to replace decay conditions due to the lack of dissipative mechanisms in equations. It is found that the turbulence and dissipation associated with wave breaking can significantly alter the flow field everywhere, not just in the breaking region. The modifications to the regime diagram caused by vertical shear and vertical variations in static stability are also elaborated.

The moments of inertia of Mars

Abstract: The mean moment of inertia of Mars is, at present, very poorly constrained. The generally accepted value of 0.365 M(R-squared) is obtained by assuming that the observed second degree gravity field can be decomposed into a hydrostatic oblate spheroid and a nonhydrostatic prolate spheroid with an equatorial axis of symmetry. An alternative decomposition is advocated in the present analysis. If the nonhydrostatic component is a maximally triaxial ellipsoid (intermediate moment exactly midway between greatest and least), the hydrostatic component is consistent with a mean moment of 0.345 M(R-squared). The plausibility of this decomposition is supported by statistical arguments and comparison with the earth, moon and Venus.

On the Pressure Gradient Force Error in σ-Coordinate Spectral Models

Abstract: The pressure gradient force error of the spectral technique used in combination with the σ vertical coordinate was examined in an idealized case of an atmosphere at rest and in hydrostatic equilibrium. Small-scale (one-point and three-point) mountains were used in the tests. With such definitions of topography, difficulties could be expected with the spectral method due to slow convergence of the Fourier series. For reference, the finite-difference pressure gradient force errors were also computed. In the examples considered, it. was found that the errors of the spectral method can be large. In the rms sense, the spectral pressure gradient force errors were larger than those of the finite-difference method.

Effects of deviations from hydrostatic equilibrium on atmospheric corrections to satellite and lunar laser range measurements

Abstract: The accuracy of the atmospheric correction to laser range measurements is one of the main limiting factors for distance measurements to retroreflectors on artificial satellites and on the Moon. The main part of this correction is currently obtained by using the local atmospheric pressure and zenith angle to determine the integrated atmospheric density along the observing path. One of the factors affecting this method of correction is possible deviations of the atmosphere from hydrostatic equilibrium for observation sites in mountainous areas. This article examines the integrated equations of motion for the atmosphere and estimates the effect of such deviations on the overall range error. It is found that the expected error is less than 1 cm most of the time, even for elevation angles as low as 20°.

Seasonal variability in the coastal circulation on the Brazilian continental shelf (29°S–35°S)

Abstract: An analytical model developed to determine coastal currents is applied to the coast of Rio Grande do Sul, Brazil (between 29°S and 35°S latitude). The data used in this study were obtained from 215 oceanographic stations in that region. Since these data were obtained on different dates, they are heterogeneous, leading us to classify, analyse and compare them on a seasonal basis. The model describes the steady-state non-tidal coastal current. The momentum balance considered is stationary and hydrostatic. Longshore pressure gradients are neglected, whilst the Coriolis parameter, f , and the vertical momentum exchange coefficient, k , are assumed constant. The mean density of the water, p, is assumed constant in the vertical, but is allowed to change at a constant rate across the shelf. The governing equation has an analytical solution obtained, by applying the boundary conditions such that: (i) the surface velocity shear is proportional to the surface wind stress; (ii) the velocity diminishes to zero at the bottom boundary. In addition, the condition of a non-divergent fluid is considered, in order to derive the sea surface slope as a function of the wind stress and the cross-shelf density gradient. Therefore, the solution establishes a single relationship between wind stress, cross-shelf mean density gradient and the fluid velocity. The density, wind and resulting velocity fields are analysed and, as a consequence, a general pattern for the coastal surface circulation system is proposed for the region.

Lyapunov stability of ideal stratified fluid equilibria in hydrostatic balance

Abstract: The inviscid dynamics of a continuously stratified, rotating, incompressible fluid in three dimensions is treated in the Boussinesq approximation, assuming hydrostatic balance in the direction of gravity. The dynamics and Lyapunov stability properties of such a fluid are studied by formulating the problem as a noncanonical Hamiltonian system in isopycnal coordinates and using Lyapunov's direct method to establish sufficient conditions for stability of a wide class of equilibrium solutions.

Hydrostatic-mechanical power-distribution transmission

Abstract: A hydrostatic-mechanical power-distribution transmission with infinitely variable gear ratio in the hydrostatic branch and several ratios in the mechanical branch, which downstream of the coupling gear has shiftable planetary reduction gears for the purpose of reducing the power contribution of the hydrostatic branch in the starting range.

Propagating and nonpropagating compression waves in an isothermal atmosphere with uniform horizontal magnetic field

Abstract: Full analytical solutions to the wave equations for steady vertical compression waves in an isothermal hydrostatic atmosphere with a uniform horizontal magnetic field are presented. It is shown that, in the steady state approach, the behavior of upward waves and downward waves is very different. It is shown that the finding of Thomas (1983), indicating that the cutoff frequency for vertically propagating magnetoacoustic waves in an isothermal atmosphere with a horizontal magnetic field is the same for isothermal atmosphere with no magnetic field, is true only for the downward waves.

Vehicle with stepless hydrostatic mechanical power branch drive - has four shaft planetary differential and two sun gears meshing respectively with planet gears

Abstract: Two main shafts (10,11) are provided to give input or output and are respectively connected to different shafts of the planet differential (4). At least two hydrostatic machines are provided, which respectively are connected in at least one working section of the power branch gear unit (1). The effective driving power introduction point (26) of the drive side main shaft (10) lies spatially between the sun gear (27) arranged at a non-drive shaft (11) and the drive engine (2) arranged in front of the power branch gear unit (1). An energy storge flywheel (16) can be coupled as required to the shaft (11) by a clutch (KS). The sun gear (27) arranged at the non drive side shaft (11) is designed with a dia. as small as possible and the planet gears (28,29) are arranged rotation proof on a shaft (30) rotationally located in the web (31). USE/ADVANTAGE - Drive unit for vehicle. Further developed to give more compact unit whilst maintaining drive principle.

Low-temperature transonic cooling flows in galaxy clusters

Abstract: Calculations are presented which demonstrate that cooling flow models with large sonic radii may be consistent with observed cluster gas properties. It is found that plausible cluster parameters and cooling flow mass accretion rates can produce sonic radii of 10-20 kpc for sonic point temperatures of 1-3 x 10 to the 6th K. The numerical calculations match these cooling flows to hydrostatic atmosphere solutions for the cluster gas beyond the cooling flow region. The cooling flows produce no appreciable 'holes' in the surface brightness toward the cluster center, and the model can be made to match the observed X-ray surface brightness of three clusters in which cooling flows had been believed to be absent. It is suggested that clusters with low velocity dispersion may be the natural location for such 'cool' cooling flows, and fits of these models to the X-ray surface brightness profiles for three clusters are presented.

Geostrophic pumping, inflows and upwelling in barrier reefs

Abstract: The communication between shallow and deep oceans via gaps in the separating barrier reefs is examined using a simplified two-layer analytical model. Attention is focused on the flow resulting from a sea-level difference between the ocean and the lagoon. Such a difference imposes a pressure gradient along the gap which, in turn, forces a flow into the lagoon. The coral reefs, which extend all the way to the surface and are exposed to the atmosphere at low tide, are presented by two portions of an infinitely long wall. A group of passages, whose combined width is not very small compared to the Rossby radius, is represented by a single gap separating the two portions of the wall. The fully nonlinear model is inviscid, hydrostatic and nondiffusive. Nonlinearity is essential because (i) the flow in the passages is rather fast, and (ii) the depth variations are of order unity. Steady solutions for the upstream and downstream fields are constructed analytically using the momentum equation in an integra...

Unsteady density-current equations for highly curved terrain

Abstract: New nonlinear partial differential equations containing terrain curvature and its rate of change are derived that describe the flow of an atmospheric density current. Unlike the classical hydraulic-type equations for density currents, the new equations are valid for two-dimensional, gradually varied flow over highly curved terrain, hence suitable for computing unsteady (or steady) flows over arbitrary mountain/valley profiles. The model assumes the atmosphere above the density current exerts a known arbitrary variable pressure upon the unknown interface. Later this is specialized to the varying hydrostatic pressure of the atmosphere above. The new equations yield the variable velocity distribution, the interface position, and the pressure distribution that contains a centrifugal component, often significantly larger than its hydrostatic component. These partial differential equations are hyperbolic, and the characteristic equations and characteristic directions are derived. Using these to form a characteristic mesh, a hypothetical unsteady curved-flow problem is calculated, not based upon observed data, merely as an example to illustrate the simplicity of their application to unsteady flows over mountains.

Hydrostatic bearing supporting device for axial piston machine

Abstract: PURPOSE:To keep off any abnormal abrasion on a pad sliding surface by feeding a hydrostatic bearing with pump discharge pressure being controlled synchronously with the tilting action of a cylinder block. CONSTITUTION:When a cylinder block 3 is 0 deg. in its tilting angle, a contraction area comes to max. and when it is a max. tilting angle, a proper contraction area is formed there. With this proper area, discharge pressure of a pump is adjusted to a proper value. Since this controlled high pressure oil is directly fed to a hydrostatic bearing 17 for supporting a piston reaction, such a piston reaction as inducing an unstable vibration in a drive disk 7 and unbalance as to thrust force of the hydrostatic bearing 17 are all in no case produced. With this constitution, the piston reaction force can be stably supported by the hydrostatic bearing 17.

Invasion of a porous medium under gravity: A quantitative analysis

Abstract: The structure of a Wood's metal [1] slow invasion front into homogeneous crushed glass is analyzed in the framework of a “gradient percolation” model allowing to account for the influence of gravity. We show that this model accounts quantitatively well for experimental observations and permits a detailed analysis of the experimental front structures. Invasion of a porous medium by a non-wetting fluid at low velocity and under zero gravity is correctly described by the “invasion percolation” model [2]. However, in most 3D systems, it is not possible to neglect the influence of gravity. The hydrostatic component adds up with the applied injection pressure: this creates a vertical gradient of the effective injection pressure. This missing gradient term has been recently introduced to analyse 2D and 3D diffusion processes [3]. In the “gradient percolation” model a linear variation of the percolation parameter is introduced along a lattice axis. As a consequence there appears a “front” connected to the high occupation region by a continuous chain of occupied sites. This “front” is equivalent to the front surface limiting the invaded volume in the invasion experiment [4]. Experiments can be shortly described as follows [1]: Wood's metal is injected at the bottom of a vertical evacuated crushed glass column. The flow velocity is kept low (a few mm/h) so that viscous pressure losses can be neglected. When the front has reached a given height, the injection is stopped and the liquid is solidified; then horizontal sections of the front corresponding to various heights z are analyzed. The pictures are digitized into a square lattice of pixels and “invaded” or “empty” pixels are discriminated by a threshold procedure. The correlation function C ( r ) of the metal distribution in horizontal planes is determined and compared to the corresponding quantity obtained from the numerical simulation for sites located on the gradient front: it is the key tool for comparing experimental and theoretical data. In both cases, close to p c in a range of r values between the individual grain size d and an upper limit γ (crossover length), C ( r ) varies roughly as: C ( r ) ∝ r D fr −2 indicating that the cut structure is fractal with a dimension D fr at short distances. When r is larger than γ, C ( r ) is constant and equal to a value S proportional to the mean metal saturation. We show that the experimental 2D correlation function C ( r ) follows the universal behaviour predicted by the gradient percolation model. A good fit is obtained for realistic values of the geometric adjustable parameters; however the value D fr = 2.4 obtained for the fractal dimension of the front is smaller than the value 2.54 expected for classical 3D percolation. The variations of the mean fluid saturation and crossover length γ with the cut height are in agreement with the numerical simulation results.

Comments on ``A One-Level Mesoscale Model for Diagnosing Surface Winds in Mountainous and Coastal Regions''

Abstract: Mass and Dempsey (1985) have reformulated the equations for the one-level sigma-system model over complex terrain and suggested they have removed the large hydrostatic part of the horizontal pressure-gradient force (PGF). It is shown that although the coefficient of g∇σZs was greatly reduced (compared to 1), the complete elimination of the hydrostatic PGF cannot be done unless the surface temperature is redefined as the sum of a perturbation temperature and a basic rest-state temperature. The alternative formula in which the hydrostatic PGF terms were totally removed is presented.

Dynamic Performance of the Stepped Hydrostatic Circumferential Gas Seal

Abstract: Theoretical analysis of dynamic tracking and leakage characteristics of a self-centering, hydrostatic, stepped gas seal is presented. The equations of motion are solved numerically, taking into account shaft speed and whirl, pressure ratio across the seal, fluid film properties and face friction effects. The analysis is limtied to low axial Reynolds numbers and negligible tangential hydrodynamic pressure buildup. Became the model neglects azimuthal pressure gradients, the analyzed seals must be either axially short or limited to small eccentricities. An analytical tracking criterion is established for high pressure ratios for preliminary design purposes. Presented at the 43rd Annual Meeting in Cleveland, Ohio May 9–12, 1988

A terrain-following coordinate system—Derivation of diagnostic relationships

Abstract: Generalized hydrostatic and geostrophic equations can be derived from the equations in the terrain-following framework. The generalized hydrostatic equation permits some non-hydrostatic motions (as obtained from a Cartesian framework) to remain when a non-zero slope exists. Correspondingly, the generalized geostrophic wind permits a horizontal divergent component (in addition to divergence caused by the change of Coriolis parameter with latitude) to occur when the slope angle is not zero.

Coordinate System-Derivation of Diagnostic Relationships

Abstract: Summary Generalized hydrostatic and geostrophic equations can be derived from the equations in the terrain-following framework. The generalized hydrostatic equation permits some non-hydrostatic motions (as obtained from a Cartesian framework) to remain when a non-zero slope exists. Correspondingly, the generalized geostrophic wind permits a horizontal divergent component (in addition to divergence caused by the change of Coriolis parameter with latitude) to occur when the slope angle is not zero.

Approximate Solution of Third-Order Clairaut Theory of Rotational Sectorial Figures of Jupiter and Saturn

Abstract: The aim of this paper is to investigate numerical solutions of third-order Clairaut theory, under the boundary conditions given in our previous work (El-Shaarawy, 1974). This solution gives an explicit form of the shape and rotational distortion, due to third-order sectorial harmonic terms, of the equipotential surfaces of the two rapidly rotating planets, Jupiter and Saturn at the different levels inside these planets owing to a certain internal density distribution model (Zharkov, 1975). We considered each of them as a heterogeneous self-gravitating fluid mass in hydrostatic equilibrium.

Planetary oblateness and evolution of Earth flattening

Abstract: SUMMARY A simple model of superficial hydrostatic equilibrium on a rotating isotropic self-gravitating sphere is employed to derive a planetary oblateness formula. The relation obtained describes oblateness on the basis of spin rate, polar radius and the mass of the planet only. An application to the oblate planets is made and the consistency with the most basic and known oblateness laws of geodesy is verified. By making the further hypothesis that the Earth volume and mass has stayed constant, one derives that the size of the terrestrial oblateness adaptation which has taken place during the last half a billion years is a contraction of 2 km at the equator an uplift of 4.1 km at the pole. For the more remote history it appears very likely that the central condensation of the Earth has decreased systematically over the past 3.5 billion years.