Showing papers on "Hydrostatic equilibrium published in 2007"

Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics

Abstract: In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics. 1. Introduction Large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations, with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, and due to the shallowness of the oceans and the atmosphere, i.e., the depth of the fluid layer is very small in comparison to the radius of the earth, the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by the hydrostatic balance. As a result

Atmospheric Dynamics of Short-period Extra Solar Gas Giant Planets I: Dependence of Night-Side Temperature on Opacity

Abstract: More than two dozen short-period Jupiter-mass gas giant planets have been discovered around nearby solar-type stars in recent years, several of which undergo transits, making them ideal for the detection and characterization of their atmospheres. Here we adopt a three-dimensional radiative hydrodynamical numerical scheme to simulate atmospheric circulation on close-in gas giant planets. In contrast to the conventional GCM and shallow water algorithms, this method does not assume quasi hydrostatic equilibrium and it approximates radiation transfer from optically thin to thick regions with flux-limited diffusion. In the first paper of this series, we consider synchronously-spinning gas giants. We show that a full three-dimensional treatment, coupled with rotationally modified flows and an accurate treatment of radiation, yields a clear temperature transition at the terminator. Based on a series of numerical simulations with varying opacities, we show that the night-side temperature is a strong indicator of the opacity of the planetary atmosphere. Planetary atmospheres that maintain large, interstellar opacities will exhibit large day-night temperature differences, while planets with reduced atmospheric opacities due to extensive grain growth and sedimentation will exhibit much more uniform temperatures throughout their photosphere's. In addition to numerical results, we present a four-zone analytic approximation to explain this dependence.

Numerical modeling of self-channeling granular flows and of their levee-channel deposits

Abstract: When not laterally confined in valleys, pyroclastic flows create their own channel along the slope by selecting a given flowing width. Furthermore, the lobe-shaped deposits display a very specific morphology with high parallel lateral levees. A numerical model based on Saint Venant equations and the empirical variable friction coefficient proposed by Pouliquen and Forterre (2002) is used to simulate unconfined granular flow over an inclined plane with a constant supply. Numerical simulations successfully reproduce the self-channeling of the granular lobe and the levee-channel morphology in the deposits without having to take into account mixture concepts or polydispersity. Numerical simulations suggest that the quasi-static shoulders bordering the flow are created behind the front of the granular material by the rotation of the velocity field due to the balance between gravity, the two-dimensional pressure gradient, and friction. For a simplified hydrostatic model, competition between the decreasing friction coefficient and increasing surface gradient as the thickness decreases seems to play a key role in the dynamics of unconfined flows. The description of the other disregarded components of the stress tensor would be expected to change the balance of forces. The front's shape appears to be constant during propagation. The width of the flowing channel and the velocity of the material within it are almost steady and uniform. Numerical results suggest that measurement of the width and thickness of the central channel morphology in deposits in the field provides an estimate of the velocity and thickness during emplacement. Citation: Mangeney, A., F. Bouchut, N. Thomas, J. P. Vilotte, and M. O. Bristeau (2007), Numerical modeling of self-channeling granular flows and of their levee-channel deposits,

AGN obscuring tori supported by infrared radiation pressure

Abstract: Explicit 2D axisymmetric solutions are found to the hydrostatic equilibrium, energy balance, and photon diffusion equations within obscuring tori around active galactic nuclei (AGNs). These solutions demonstrate that infrared radiation pressure can support geometrically thick structures in AGN environments subject to certain constraints: the bolometric luminosity must be roughly ~0.03-1 times the Eddington luminosity; and the Compton optical depth of matter in the equatorial plane should be ~1, with a tolerance of about an order of magnitude up or down. Both of these constraints are at least roughly consistent with observations. In addition, angular momentum must be redistributed so that the fractional rotational support against gravity rises from the inner edge of the torus to the outer in a manner specific to the detailed shape of the gravitational potential. This model also predicts that the column densities observed in obscured AGNs should range from ~1022 to ~1024 cm-2.

The Hot Interstellar Medium of Normal Elliptical Galaxies. I. A Chandra Gas Gallery and Comparison of X-Ray and Optical Morphology

Abstract: We present an X-ray analysis of 54 normal elliptical galaxies in the Chandra archive and isolate their hot gas component from the contaminating point-source emission, allowing us to conduct, for the first time, a morphological analysis on the gas alone. A comparison with optical images and photometry shows that the hot gas morphology has surprisingly little in common with the shape of the stellar distribution. We observe no correlation between optical and X-ray ellipticities in the inner regions where stellar mass dominates over dark matter. A shallow correlation would be expected if the gas had settled into hydrostatic equilibrium with the gravitational potential. Instead, observed X-ray ellipticities exceed optical ellipticities in many cases. We exclude rotation as the dominant factor to produce the gas ellipticities. The gas appears disturbed, and hydrostatic equilibrium is the exception rather than the rule. Nearly all hydrostatic models can be ruled out at 99% confidence, based on their inability to reproduce the optical-X-ray correlation and large X-ray ellipticities. Hydrostatic models not excluded are those in which dark matter either dominates over stellar mass inside the inner half-light radius or has a prominently cigar-shaped distribution, both of which can be ruled out on other grounds. We conclude that, even for rather X-ray-faint elliptical galaxies, the gas is at least so far out of equilibrium that it does not retain any information about the shape of the potential, and that X-ray-derived radial mass profiles may be in error by factors of order unity.

A Theoretical and Numerical Study of Urban Heat Island-Induced Circulation and Convection

Abstract: Urban heat island–induced circulation and convection in three dimensions are investigated theoretically and numerically in the context of the response of a stably stratified uniform flow to specified low-level heating that represents an urban heat island. In a linear, theoretical part of the investigation, an analytic solution for the perturbation vertical velocity in a three-dimensional, time-dependent, hydrostatic, nonrotating, inviscid, Boussinesq airflow system is obtained. The solution reveals a typical internal gravity wave field, including low-level upward motion downwind of the heating center. Precipitation enhancement observed downwind of urban areas may be partly due to this downwind upward motion. The comparison of two- and three-dimensional flow fields indicates that the dispersion of gravity wave energy into an additional dimension results in a faster approach to a quasi-steady state and a weaker quasi-steady flow well above the concentrated heating region in three dimensions. In a n...

On the Hydrostatic Theory of the Figure of the Earth

Abstract: Summary A table is provided of the external and dynamical ellipticities of the Earth for different values of the ratio I/Mb2, I being the mean moment of inertia and b the mean radius, on the first order Radau theory. A simplified model is used to find the corrections for the slight inaccuracy of the Radau approximation and the terms of the second order. The dynamical ellipticity and J2 as given by artificial satellites are used to estimate I/Mb2, and it is shown that on the hydrostatic theory 106J2 would be 1072.1 ± 0.4, in contradiction to the direct determination 1082.78 ± 0.05. The departures from the hydrostatic state indicated by J2 and J3 imply stress differences of 4 × 107 to 8 × 107 dyn/cm2 in the interior.

Internal Tide Generation at the Continental Shelf Modeled Using a Modal Decomposition: Two-Dimensional Results

Abstract: Stratified flow over topography is studied, with oceanic applications in mind. A model is developed for a fluid with arbitrary vertical stratification and a free surface, flowing over three-dimensional topography of arbitrary size and steepness, with background rotation, in the linear hydrostatic regime. The model uses an expansion of the flow fields in terms of a set of basis functions, which efficiently capture the vertical dependence of the flow. The horizontal structure may then be found by solving a set of coupled partial differential equations in two horizontal directions and time, subject to simple boundary conditions. In some cases, these equations may be solved analytically, but, in general, simple numerical procedures are required. Using this formulation, the internal tide generated by a time-periodic barotropic tidal flow over a continental shelf and slope is calculated in various idealized configurations. The topography and fluid motion are taken to be independent of one coordinate di...

Mathematical Justification of a Shallow Water Model

Abstract: The shallow water equations are widely used to model the flow of a thin layer of fluid submitted to gravity forces. They are usually formally derived from the full incompressible Navier-Stokes equations with free surface under the modeling hypothesis that the pressure is hydrostatic, the flow is laminar, gradually varied and the characteristic fluid height is small relative to the characteristics flow length. This paper deals with the mathematical justification of such asymptotic process assuming a non zero surface tension coefficient and some constraints on the data. We also discuss relation between lubrication models and shallow water systems with no surface tension coefficient necessity.

A note on the slim accretion disk model

Abstract: We show that when the gravitational force is correctly calculated in dealing with the vertical hydrostatic equilibrium of black hole accretion disks, the relationship that is valid for geometrically thin disks, i.e., cs/ΩKH = constant, where cs is the sound speed, ΩK is the Keplerian angular velocity, and H is the half-thickness of the disk, does not hold for slim disks. More importantly, by adopting the correct vertical gravitational force in studies of thermal equilibrium solutions, we find that there exists a maximal possible accretion rate for each radius in the outer region of optically thick accretion flows, such that only the inner regions of these flows can possibly take the form of slim disks, and strong outflows from the outer region are required to reduce the accretion rate in order for slim disks to be realized.

Stiffness and Efficiency Optimization of a Hydrostatic Laser Surface Textured Gas Seal

Abstract: Microdimples generated by laser surface texturing (LST) can be used to enhance performance in hydrostatic gas-lubricated mechanical seals. This is achieved by applying microdimples with high area density over a certain portion of the sealing dam width adjacent to the high-pressure side, leaving the remaining portion untextured. The textured portion provides an equivalent larger gap that results in converging clearance in the direction of pressure drop and hence, hydrostatic pressure buildup, similar to that of a radial step seal. A mathematical model based on the solution of the Reynolds equation for compressible Newtonian fluid in a narrow gap between two nominally parallel stationary surfaces is developed. A detailed dimensionless analysis of the texturing parameters is performed to achieve maximum gas film stiffness with minimum gas leakage. DOI: 10.1115/1.2540120

Hydrostatic equilibrium equation and Newtonian limit of the singular f(R) gravity

Abstract: We derive the hydrostatic equilibrium equation of a spherical star for any gravitational Lagrangian density of the form . The Palatini variational principle for the Helmholtz Lagrangian in the Einstein gauge is used to obtain the field equations in this gauge. The equilibrium hydrostatic equation is obtained and is used to study the Newtonian limit for . The same procedure is carried out for the more generally case giving a good Newtonian limit.

Equivalent Stiffness and Damping Investigation of a Hydrostatic Journal Bearing

Abstract: The aim of this research is to study the dynamic characteristics of a hydrostatic journal bearing, with four hydrostatic bearing flat pads fed by diaphragm restrictors and supporting a rotor. We assumed that the fluid flow is incompressible, laminar, isothermal, and steady-state. Linear modeling was performed using a numerical method in order to investigate the effects of the film thickness, the recess pressure, and the geometric configuration on the equivalent stiffness and damping of a hydrostatic journal bearing. In the first step, the variation of equivalent stiffness and damping is studied according to the pressure ratio for different geometric configurations of a hydrostatic bearing at the point of operation. In the second step, the variation of the equivalent stiffness and damping is studied according to the ratio of the film thickness for different geometric configurations of a hydrostatic bearing when one moves away from the point of operation. The results show that the hydrostatic journal bearin...

The formation of HD 149026 b

Abstract: Today, many extrasolar planets have been detected. Some of them exhibit properties quite different from the planets in our Solar system and they have eluded attempts to explain their formation. One such case is HD 149026 b. It was discovered by Sato et al. A transit-determined orbital inclination results in a total mass of . The unusually small radius can be explained by a condensible element core with an inferred mass of for the best-fitting theoretical model. In the core accretion model, giant planets are assumed to form around a growing core of condensible materials. With increasing core mass, the amount of gravitationally bound envelope mass increases. This continues up to the so-called critical core mass – the largest core allowing a hydrostatic envelope. For larger cores, the lack of static solutions forces a dynamic evolution of the protoplanet, accreting large amounts of gas or ejecting the envelope in the process. This would prevent the formation of HD 149026 b. By studying all possible hydrostatic equilibria we could show that HD 149026 b can remain hydrostatic up to the inferred heavy core. This is possible if it is formed in situ in a relatively low-pressure nebula. This formation process is confirmed by fluid-dynamic calculations using the environmental conditions as determined by the hydrostatic models. We present a quantitative in situ formation scenario for the massive core planet HD 149026 b. Furthermore, we predict a wide range of possible core masses for close-in planets like HD 149026 b. This is different from migration, where typical critical core masses should be expected.

Linear Anelastic Equations for Atmospheric Vortices

Abstract: A linear anelastic-vortex model is derived using assumptions appropriate to waves on vortices with scales similar to tropical cyclones. The equation set is derived through application of a multiple-scaling technique, such that the radial variations of the thermodynamic fields are incorporated into the reference state. The primary assumption required for the model is that the horizontal variations in the thermodynamic variables describing the reference state are appreciably longer than the waves on the vortex. This new version of the anelastic system makes no approximation to the requirements for hydrostatic and gradient wind balance, or the buoyancy frequency, in the core of the vortex. A small but measurable improvement in the performance of the new equation set is demonstrated through simulations of gravity waves and vortex–Rossby waves in a baroclinic vortex.

Hydrodynamic simulations of irradiated secondaries in dwarf novae

Abstract: Context. Secondary stars in dwarf novae are strongly irradiated during outbursts. It has been argued that this could result in an enhancement of the mass transfer rate even though the L 1 region is shaded from the primary irradiation by the accretion disc. Previous investigations of the possibility of a circulation flow transporting heat from hot regions to L 1 have given opposite answers. Aims. We investigate the surface flow of irradiated secondaries numerically. We consider the full time-dependent problem and take the two-dimensional nature of the flow into account. Methods. We use a simple model for the irradiation and the geometry of the secondary star: the irradiation temperature is treated as a free parameter and the secondary is replaced by a spherical star with a space-dependent Coriolis force that mimics the effect of the Roche geometry. The Euler equations are solved in spherical coordinates with the TVD-MacCormack scheme. Results. We show that the Coriolis force leads to the formation of a circulation flow from the high-latitude region to the close vicinity of the L 1 point. However, no heat can be efficiently transported to the L 1 region due to the rapid radiative cooling of the hot material as it enters the equatorial belt shaded from irradiation. Under the assumption of hydrostatic equilibrium, the Coriolis force could lead to a moderate increase in the mass transfer by pushing the gas in the vertical direction in the vicinity of L 1 , but only during the initial phases of the outburst (about 15-20 orbital periods). It remains possible, however, that this assumption breaks up due to the strong surface velocity of the flow transiting by L 1 , close to the sound speed. In this case, however, a three-dimensional approach would then be needed to determine the mass flux leaving the secondary. Conclusions. We therefore conclude that the Coriolis force does not prevent a flow from the heated regions of the secondary towards the L 1 region, at least during the initial phases of an outburst, but the resulting increase in the mass transfer rate is moderate, so unlikely to be able to account for the duration of long outbursts.

Exact Solutions of the Isothermal Lane--Emden Equation with Rotation and Implications for the Formation of Planets and Satellites

Abstract: We have derived exact solutions of the isothermal Lane--Emden equation with and without rotation in a cylindrical geometry. The corresponding hydrostatic equilibria are most relevant to the dynamics of the protosolar nebula before and during the stages of planet and satellite formation. The nonrotating solution for the mass density is analytic, nonsingular, monotonically decreasing with radius, and it satisfies easily the usual physical boundary conditions at the center. When differential rotation is added to the Lane--Emden equation, an entire class of exact solutions for the mass density appears. We have determined all of these solutions analytically as well. Within this class, solutions that are power laws or combinations of power laws are not capable of satisfying the associated boundary--value problem, but they are nonetheless of profound importance because they constitute "baselines" to which the actual solutions approach when the central boundary conditions are imposed. Numerical integrations that enforce such physical boundary conditions show that the actual radial equilibrium density profiles emerge from the center close to the nonrotating solution, but once they cross below the corresponding baselines, they cease to be monotonic. The actual solutions are forced to oscillate permanently about the baseline solutions without ever settling onto them because the central boundary conditions strictly prohibit such settling, even in the asymptotic regime of large radii. Based on our results, we expect that quasistatically--evolving protoplanetary disks should develop oscillatory density profiles in their midplanes during their isothermal phase. The peaks in these profiles correspond to local potential minima and their locations are ideal sites for the formation of protoplanets ...

Probing the dynamical state of galaxy clusters

Abstract: We show how hydrostatic equilibrium in galaxy clusters can be quantitatively probed combining X-ray, SZ, and gravitational-lensing data. Our previously published method for recovering three-dimensional cluster gas distributions avoids the assumption of hydrostatic equilibrium. Independent reconstructions of cumulative total-mass profiles can then be obtained from the gas distribution, assuming hydrostatic equilibrium, and from gravitational lensing, neglecting it. Hydrostatic equilibrium can then be quantified comparing the two. We describe this procedure in detail and show that it performs well on progressively realistic synthetic data. An application to a cluster merger demonstrates how hydrostatic equilibrium is violated and restored as the merger proceeds.

Comparing analytical and numerical solution of nonlinear two and three‐dimensional hydrostatic flows

Abstract: New test cases for frictionless, three-dimensional hydrostatic flows have been derived from some known analytical solutions of the two-dimensional shallow water equations. The flow domain is a paraboloid of revolution and the flow is determined by the initial conditions, the nonlinear advective terms, the Coriolis acceleration and by the hydrostatic pressure. Wetting and drying is also included. Some specific properties of the exact solutions are discussed under different hypothesis and relative importance of the forcing terms. These solutions are proposed for testing the stability, the accuracy and the efficiency of numerical models to be used for simulating environmental hydrostatic flows. The computed solutions obtained with a semi-implicit finite difference-finite volume algorithm on unstructured grid are compared with the corresponding analytical solutions in both two and three space dimension. Excellent agreement are obtained for the velocity and for the resulting water surface elevation. Comparison of the computed inundation area also shows a good agreement with the analytical solution with degrading accuracy observed when the inundation area becomes relatively large and for long simulation time.

Simulation of dynamic effects on hydrostatic bearings and membrane restrictors

Abstract: Hydrostatic bearings have an excellent static and dynamic behavior and are used for different kinds of application. Application of hydrostatic bearings is limited by friction and therewith by velocity. Typical characteristics of the hydrostatic system (load, stiffness, flow) are calculated without a velocity dependency. The geometry of the hydrostatic bearing pockets and their restrictors are optimized by using time continuous pressure distribution at the bearing pocket, laminar flow behavior as well as constant velocity of the bearing. The dynamic effects of the flow at high velocities are not considered. The paper reflects the common design and calculation methods and shows their limitations in regard to the calculation of hydrostatic bearings at high velocities. It analyzes the results of complex dynamic flow simulations of hydrostatic bearings and presents a new design and optimization concept of hydrostatic bearings. This concept analyses the oil flow at high bearing velocities and it optimizes the bearing geometry, the restrictor geometry as well as the geometry of the main mechanical components.

Probing the dynamical state of galaxy clusters

Abstract: We show how hydrostatic equilibrium in galaxy clusters can be quantitatively probed combining X-ray, SZ, and gravitational-lensing data. Our previously published method for recovering three-dimensional cluster gas distributions avoids the assumption of hydrostatic equilibrium. Independent reconstructions of cumulative total-mass profiles can then be obtained from the gas distribution, assuming hydrostatic equilibrium, and from gravitational lensing, neglecting it. Hydrostatic equilibrium can then be quantified comparing the two. We describe this procedure in detail and show that it performs well on progressively realistic synthetic data. An application to a cluster merger demonstrates how hydrostatic equilibrium is violated and restored as the merger proceeds.

General relativistic effects of strong magnetic fields on the gravitational force: a driving engine for bursts of gamma rays in SGRs?

Abstract: In general relativity all forms of energy contribute to gravity and not only just ordinary matter as in Newtonian Physics. This fact can be seen in the modified hydrostatic equilibrium equation for relativistic stars pervaded by magnetic (B) fields. It has an additional term coupled to the matter part as well as an anisotropic term which is purely of magnetic origin. That additional term coming from the pressure changed by the radial component of the diagonal electromagnetic field tensor, weakens the gravitational force when B is strong enough and can even produce an unexpected change in the attractive nature of the force by reversing its sign. In an extreme case, this new general relativistic (GR) effect can even trigger an instability in the star as a consequence of the sudden reversal of the hydrostatic pressure gradient. We suggest here that this GR effect may be the possible central engine driving the transient giant outbursts observed in Soft Gamma-ray Repeaters (SGRs). In small regions of the neutron star (NS), strong magnetic condensation can take place. Beyond a critical limit, these highly magnetised bubbles may explode releasing the trapped energy as a burst of -rays of � 10 36 40 erg.

On the influence of fluid-structure-interaction on the stability of thin-walled shell structures

Abstract: The purpose of this contribution is to investigate the stabilizing influence of hydrostatic fluid and/or gas pressure supports (especially the effect of the volume dependency on gas or fluid pressures) on the stability, namely the eigenvalues and eigenmodes of the stiffness matrix of shell or membrane-like structures undergoing large displacements. For this purpose, an analytical mesh free or lumped parameter description for the fluid/gas (see also Refs. 3, 12, 13 and 14) is taken, which yields a special structure of the nonlinear equations representing the change of the gas or fluid volume or alternatively the change of the wetted part of the shell surface. Finally this procedure leads to a stiffness matrix with several rank updates — depending on the volume containing either gas or fluid or both. These rank updates are a key part in the stability analysis: they describe the different coupling of the fluid or gas volume change with the structural displacements in addition to the deformation dependence of the standard pressure. The specific rank updates allow the derivation of a very efficient algorithm to compute the modifications of the eigenvalues and eigenmodes of the original stiffness matrix without gas or fluid loading or support.

Hydrostatic Experiments up to Ultrahigh Pressures

Abstract: The development of the diamond-anvil-cell technology has enabled us to conduct in situ experiments at ultrahigh pressures. The current pressure limit reaches well beyond 200 GPa. The stress environment in such experiments is in general highly nonhydrostatic, and one has to be careful in evaluating the obtained data. Nonhydrostaticity affects the phase stability, transition pressures, equation of state, and various lattice properties. For detailed discussion of the change in physical properties with pressure, we have to realize hydrostatic or quasihydrostatic conditions. I describe our present understanding of the stress state of a sample compressed in a diamond-anvil cell, together with recent efforts for extending the pressure limit of hydrostatic experiments.