Showing papers on "Hydrostatic equilibrium published in 2006"

Global MHD simulations of stratified and turbulent protoplanetary discs - I. Model properties

Abstract: Aims. We present the results of global 3-D MHD simulations of stratified and turbulent protoplanetary disc models. The aim of this work is to develop thin disc models capable of sustaining turbulence for long run times, which can be used for on-going studies of planet formation in turbulent discs. Methods. The results are obtained using two codes written in spherical coordinates: GLOBAL and NIRVANA. Both are time-explicit and use finite differences along with the Constrained Transport algorithm to evolve the equations of MHD. Results. In the presence of a weak toroidal magnetic field, a thin protoplanetary disc in hydrostatic equilibrium is destabilised by the magnetorotational instability (MRI). When the resolution is large enough (∼25 vertical grid cells per scale height), the entire disc settles into a turbulent quasi steady-state after about 300 orbits. Angular momentum is transported outward such that the standard a parameter is roughly 4-6 x 10 -3 . We find that the initial toroidal flux is expelled from the disc midplane and that the disc behaves essentially as a quasi-zero net flux disc for the remainder of the simulation. As in previous studies, the disc develops a dual structure composed of an MRI-driven turbulent core around its midplane, and a magnetised corona stable to the MRI near its surface. By varying disc parameters and boundary conditions, we show that these basic properties of the models are robust. Conclusions. The high resolution disc models we present in this paper achieve a quasi-steady state and sustain turbulence for hundreds of orbits. As such, they are ideally suited to the study of outstanding problems in planet formation such as disc-planet interactions and dust dynamics.

Exact Solutions of Einstein's Field Equations

Abstract: We examine various well known exact solutions available in the literature to investigate the recent criterion obtained in Negi and Durgapal [Gravitation and Cosmology 7, 37 (2001)] which should be fulfilled by any static and spherically symmetric solution in the state of hydrostatic equilibrium. It is seen that this criterion is fulfilled only by (i) the regular solutions having a vanishing surface density together with pressure, and (ii) the singular solutions corresponding to a non-vanishing density at the surface of the configuration. On the other hand, the regular solutions corresponding to a non-vanishing surface density do not fulfill this criterion. Based upon this investigation, we point out that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the state of hydrostatic equilibrium in general relativity. The regular solutions with finite centre and non-zero surface densities which do not fulfill the criterion given by Negi and Durgapal (2001), in fact, cannot meet the requirement of the‘actual mass’, set up by exterior Schwarzschild solution. The only regular solution which could be possible in this regard is represented by uniform (homogeneous) density distribution. This criterion provides a necessary and sufficient condition for any static and spherical configuration (including core-envelope models) to be compatible with the structure of general relativity [that is, the state of hydrostatic equilibrium in general relativity]. Thus, it may find application to construct the appropriate core-envelope models of stellar objects like neutron stars and may be used to test various equations of state for dense nuclear matter and the models of relativistic star clusters with arbitrary large central redshifts.

3D-radiation hydro simulations of disk-planet interactions. I. Numerical algorithm and test cases

Abstract: We study the evolution of an embedded protoplanet in a circumstellar disk using the 3D-Radiation Hydro code TRAMP, and treat the thermodynamics of the gas properly in three dimensions. The primary interest of this work lies in the demonstration and testing of the numerical method. We show how far numerical parameters can influence the simulalions of gap opening. We study a standard reference model under various numerical approximations. Then we compare the commonly used locally isothermal approximation to the radialion hydro simulation using an equation for the internal energy. Models with different treatments of the mass accretion process are compared. Often mass accumulates in the Roche lobe of the planet creating a hydrostatic atmosphere around the planet. The gravitational torques induced by the spiral pattern of the disk onto the planet are not strongly affected in the average magnitude, but the short time scale fluctuations are stronger in the radiation hydro models. An interesting result of this work lies in the analysis of the temperature structure around the planet. The most striking effect of treating the thermodynamics properly is the formation of a hot pressure-supported bubble around the planet with a pressure scale height of H/R 0.5 rather than a thin Keplerian circumplanetary accretion disk.

Structure of Magnetic Tower Jets in Stratified Atmospheres

Abstract: Based on a new approach on modeling the magnetically dominated outflows from AGNs (Li et al. 2006), we study the propagation of magnetic tower jets in gravitationally stratified atmospheres (such as a galaxy cluster environment) in large scales (> tens of kpc) by performing three-dimensional magnetohydrodynamic (MHD) simulations. We present the detailed analysis of the MHD waves, the cylindrical radial force balance, and the collimation of magnetic tower jets. As magnetic energy is injected into a small central volume over a finite amount of time, the magnetic fields expand down the backgroun d density gradient, forming a collimated jet and an expanded “lobe” due to the gradually decreasing background density and pressure. Both the jet and lobes are magnetically dominated. In addition, the injection and expansion produce a hydrodynamic shock wave that is moving ahead of and enclosing the magnetic tower jet. This shock can eventually break the hydrostatic equilibrium in the ambient medium and cause a global gravitational contraction. This contraction produces a strong compression at the head of the magnetic tower front and helps to collimate radially to produce a slendershaped jet. At the outer edge of the jet, the magnetic pressur e is balanced by the background (modified) gas pressure, without any significant contribution from the hoo p stress. On the other hand, along the central axis of the jet, hoop stress is the dominant force in shaping the cent ral collimation of the poloidal current. The system, which possesses a highly wound helical magnetic configurati on, never quite reaches a force-free equilibrium state though the evolution becomes much slower at late stages. The simulations were performed without any initial perturbations so the overall structures of the jet r emain mostly axisymmetric. Subject headings:magnetic fields — galaxies: active — galaxies: jets — methods : numerical — magnetohydrodynamics (MHD)

Rotational stability of dynamic planets with elastic lithospheres

Abstract: Received 13 April 2005; revised 21 October 2005; accepted 3 November 2005; published 14 February 2006. [1] We revisit the classic problem of the secular rotational stability of planets in response to loading using the fluid limit of viscoelastic Love number theory. Gold (1955) and Goldreich and Toomre (1969) considered the stability of a hydrostatic planet subject to an uncompensated surface mass load and concluded that a mass of any size would drive true polar wander (TPW) that ultimately reorients the load to the equator. Willemann (1984) treated the more self-consistent problem where the presence of a lithosphere leads to both imperfect load compensation and a remnant rotational bulge. Willemann considered axisymmetric loads and concluded that the equilibrium pole location was governed by a balance, independent of elastic lithospheric thickness, between the loadinduced TPW and stabilization by the remnant bulge. Our new analysis demonstrates that the equilibrium pole position is a function of the lithospheric strength, with a convergence to Willemann’s results evident at high values of elastic thickness (>400 km for an application to Mars), and significantly larger predicted TPW for planets with thin lithospheres. Furthermore, we demonstrate that nonaxisymmetric surface mass loads and internal (convective) heterogeneity, even when these are small relative to axisymmetric contributions, can profoundly influence the rotational stability. Indeed, we derive the relatively permissive conditions under which nonaxisymmetric forcing initiates an inertial interchange TPW event (i.e., a 90� pole shift). Finally, Willemann’s analysis is often cited to argue for a small (<18� ) TPW of Mars driven by the development of a Tharsissized load. We show that even in the absence of the destabilizing effects of load asymmetry, the equations governing rotational stability permit higher excursions of the Martian rotation vector than has previously been appreciated.

Hydrostatic models for the rotation of extra-planar gas in disk galaxies

Abstract: We show that fluid stationary models are able to reproduce the observed, negative vertical gradient of the rotation velocity of the extra-planar gas in spiral galaxies. We have constructed models based on the simple condition that the pressure of the medium does not depend on density alone (baroclinic instead of barotropic solutions: isodensity and isothermal surfaces do not coincide). As an illustration, we have successfully applied our method to reproduce the observed velocity gradient of the lagging gaseous halo of NGC 891. The fluid stationary models discussed here can describe a hot homogeneous medium as well as a "gas" made of discrete, cold H i clouds with an isotropic velocity dispersion distribution. Although the method presented here generates a density and velocity field consistent with observational constraints, the stability of these configurations remains an open question.

Direct Measurements of Gas Bulk Flows in the Intracluster Medium of the Centaurus Cluster with the Chandra Satellite

Abstract: We present the analysis of the velocity structure of the intracluster gas near the core of Abell 3526 obtained with two off-center Chandra observations, specifically designed to eliminate errors due to spatial variations of the instrumental gain. We detected a significant velocity gradient along the northeast-southwest direction, roughly perpendicular to the direction of the incoming subgroup Cen 45, in agreement with previous ASCA SIS measurements. The presence of gas bulk velocities is observed both with and without the inclusion of the Fe K line complex in the spectral fittings. The configuration and magnitude of the velocity gradient is consistent with near transonic circulatory motion, either bulk or eddylike. The velocity difference obtained using the best calibrated central regions of ACIS-S3 is found to be (2.4 ± 1.0) × 103 km s-1 for rectangular regions 24 × 3' roughly diametrically opposed around the cluster's core. There are also indications of a high-velocity zone toward the southern region with similar magnitudes. The detection of velocity gradients is significant at >99.4% confidence, and simulations show that intrachip gain fluctuations >1800 km s-1 are required to explain the velocity gradient by chance. The measurements suggest that >1% of the total merger energy can still be bulk kinetic 0.4 Gyr after the merging event. This is the first direct confirmation of velocity gradients in the intracluster gas with independent instruments and indicates that strong departure from hydrostatic equilibrium is possible even for cool clusters that do not show obvious signs of merging.

Magnetic Pressure Support and Accretion Disk Spectra

Abstract: Stellar atmosphere models of ionized accretion disks have generally neglected the contribution of magnetic fields to the vertical hydrostatic support, although magnetic fields are widely believed to play a critical role in the transport of angular momentum. Simulations of magnetorotational turbulence in a vertically stratified shearing box geometry show that magnetic pressure support can be dominant in the upper layers of the disk. We present calculations of accretion disk spectra that incorporate vertical magnetic pressure and dissipation profiles derived from the radiation magnetohydrodynamical simulation of Hirose, Krolik, & Stone. Magnetic pressure support generically produces a more vertically extended disk atmosphere with a larger density scale height. This acts to harden the spectrum compared to models that neglect magnetic pressure support. We estimate the significance of this effect on disk-integrated spectra by calculating an illustrative disk model for a stellar mass black hole, assuming that similar magnetic pressure support exists at all radii.

Dynamic regimes of buoyancy-affected two-phase flow in unconsolidated porous media.

Abstract: The invasion and subsequent flow of a nonwetting fluid (NWF) in a three-dimensional, unconsolidated porous medium saturated with a wetting fluid of higher density and viscosity have been studied experimentally using a light-transmission technique. Distinct dynamic regimes have been found for different relative magnitudes of viscous, capillary, and gravity forces. It is shown that the ratio of viscous and hydrostatic pressure gradients can be used as a relevant dimensionless number $K$ for the characterization of the different flow regimes. For low values of $K$, the invasion is characterized by the migration and fragmentation of isolated clusters of the NWF resulting from the prevalence of gravity and capillary forces. At high values of $K$, the dominance of viscous and gravity forces leads to an anisotropic fingerlike invasion. When the invasion stops after the breakthrough of the NWF at the open upper boundary, the invasion structure retracts under the influence of gravity and transforms into stable vertical channels. It is shown that the stability of these channels is the result of a balance between hydrostatic and viscous pressure gradients.

Forced Obliquity and Moments of Inertia of Titan

Abstract: The obliquity of Titan is small, but certainly non-zero, and may be used to place constraints on Titan's internal structure. The measured gravity coefficients of Titan imply that it is non-hydrostatic and thus the normal Darwin–Radau approach to determining internal structure cannot be applied. However, if the obliquity is assumed to be tidally damped (that is, in a Cassini state) then combining the obliquity with the measured gravity coefficients allows Titan's moment of inertia to be determined without invoking hydrostatic equilibrium. For polar moment values in the range ( 0.3 C / M R 2 0.4 ), tidally-damped obliquity values of ( 0.115 ° | e | 0.177 ° ) result. If the inferred moment value exceeds 0.4, this strongly suggests the presence of a near-surface ice shell decoupled from the interior, probably by a subsurface ocean.

Reconciling optical and X‐ray mass estimates: the case of the elliptical galaxy NGC 3379

Abstract: NGC 3379 is a well-studied nearby elliptical for which optical investigations have claimed a little dark matter content, or even no dark matter. Recently, its total mass profile M(r) has been derived by exploiting Chandra observations of its extended and X-ray emitting interstellar medium, based on the hypothesis of hydrostatic equilibrium for the hot gas. The resulting total mass within the effective radius R e has been claimed to be a few times larger than that found by optical studies. Here, we show that part of the discrepancy can be due to an underestimate of the optically derived mass, and the remaining discrepancy of a factor of ∼2 can be explained by deviations from hydrostatic equilibrium of the hot gas. By using hydrodynamical simulations tailored to reproduce the observed hot gas properties of NGC 3379, and by assuming as input for the simulations the total mass profile derived optically, we show that (i) the hot gas at the present time has X-ray properties consistent with those observed only if it is outflowing over most of the galactic body, and (ii) an overestimate of M of the same size found in the recent X-ray analysis is recovered when assuming hydrostatic equilibrium. We also show that the hot gas is outflowing even for a dark matter fraction within R e as large as derived with the standard X-ray procedure based on the hydrostatic equilibrium assumption, which shows the unapplicability of the method for this galaxy. Finally, we find that the whole range of dark mass amount and distribution allowed for by optical studies is compatible with a hot gas flow with the observed X-ray properties.

Reconciling optical and X-ray mass estimates: the case of the elliptical galaxy NGC3379

Abstract: NGC3379 is a well studied nearby elliptical for which optical investigations have claimed a little dark matter content, or even no dark matter. Recently, its total mass profile M(r) has been derived by exploiting Chandra observations of its extended and X-ray emitting interstellar medium, based on the hypothesis of hydrostatic equilibrium for the hot gas. The resulting total mass within the effective radius Re has been claimed to be a few times larger than found by optical studies. Here we show that part of the discrepancy can be due to an underestimate of the optically derived mass, and the remaining discrepancy of a factor of ~2 can be explained by deviations from hydrostatic equilibrium of the hot gas. By using hydrodynamical simulations tailored to reproduce the observed hot gas properties of NGC3379, and by assuming as input for the simulations the total mass profile derived optically, we show that i) the hot gas at the present time has X-ray properties consistent with those observed only if it is outflowing over most of the galactic body, and ii) an overestimate of M of the same size found in the recent X-ray analysis is recovered when assuming hydrostatic equilibrium. We also show that the hot gas is outflowing even for a dark matter fraction within Re as large as derived with the standard X-ray procedure based on the hydrostatic equilibrium assumption, which shows the unapplicability of the method for this galaxy. Finally, we find that the whole range of dark mass amount and distribution allowed for by optical studies is compatible with a hot gas flow with the observed X-ray properties.

Magnetic pressure support and accretion disk spectra

Abstract: Stellar atmosphere models of ionized accretion disks have generally neglected the contribution of magnetic fields to the vertical hydrostatic support, although magnetic fields are widely believed to play a critical role in the transport of angular momentum. Simulations of magnetorotational turbulence in a vertically stratified shearing box geometry show that magnetic pressure support can be dominant in the upper layers of the disk. We present calculations of accretion disk spectra that include this magnetic pressure support, as well as a vertical dissipation profile based on simulation. Magnetic pressure support generically produces a more vertically extended disk atmosphere with a larger density scale height. This acts to harden the spectrum compared to models that neglect magnetic pressure support. We estimate the significance of this effect on disk-integrated spectra by calculating an illustrative disk model for a stellar mass black hole, assuming that similar magnetic pressure support exists at all radii.

Thick-plate flexure re-examined

Abstract: The flexure of an incompressible, thick elastic plate floating on an inviscid substratum and subject to an external gravity field is re-analysed. The solution is derived from momentum equations which account for the advection of hydrostatic pre-stress. This is contrasted with a recently published thick-plate solution derived from momentum equations without a pre-stress term. It is demonstrated that neglecting pre-stress advection renders the solution singular when the model degenerates into an inviscid half-space. If pre-stress advection is included, the solution remains correct in this limit. A numerical comparison of both types of thick-plate solution with results based on conventional thin-plate theory further shows that, for geophysically relevant models, the difference in the momentum balance entails discrepancies between the thick-plate solutions which are comparable to the errors introduced by the thin-plate approximation.

A 3-D non-hydrostatic pressure model for small amplitude free surface flows

Abstract: A three-dimensional, non-hydrostatic pressure, numerical model with k-e equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non-hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non-hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure-Poisson equation is solved iteratively with a preconditioned linear GMRES method

Application of radial x-ray diffraction to determine the hydrostatic equation of state and strength of TiB2 up to 60GPa

Abstract: Room temperature investigations of the shear strength of hexagonal TiB2 have been performed in order to determine the hydrostatic equation of state of the material up to 60GPa using radial x-ray diffraction in a diamond anvil cell. We have analyzed the deformation mechanisms under pressure by analyzing the (001) and (100) peaks in the powder diffraction data, and we have deduced the hydrostatic equation of state of TiB2. The uniaxial stresses in the (100), (001), and (101) diffraction planes show a large pressure dependence, indicating a significantly large anisotropy in the material. The stress in the (001) plane shows the largest increase with pressure and reaches a maximum value before the other planes, indicating an initial activation of slip in the (001) plane at the onset of plastic deformation. Compared to gold, the averaged uniaxial stress component in TiB2 is almost 27 times as large at the maximum loading pressure, 60GPa, achieved in the experiment.

Internal Structure of Rhea

Abstract: We model the interior of Rhea on the basis of observational constraints and the results from geodynamical models available in the literature. Ten main types of models are defined, depending on the presence or absence of a high-pressure ice layer (ice II), and the extent of separation of the rock component from the volatiles. The degree-two gravity coefficients are computed for each of these models in order to assess which properties of the interior are likely to be inferred from Cassini radio science measurements scheduled on 26 November 2005. C22 greater than 2.5 x 10(exp -4) indicates that the satellite is undifferentiated, except for a slight increase in density with depth resulting from material self-compression. C22 between 1.67 x 10(exp -4)(lower bound) and 1.90 x 10(exp -4) indicates the presence of a rocky core, whose radius can be determined from the satellite's mass and ices densities, for a given temperature profile. For other values, most of the ten models cannot be distinguished from each other. However, assumptions on the density of the rock phase, presence or absence of ice II, and the degree of differentiation could allow a unique model to be determined in many cases. While the calculation presented in this work assumes that Rhea is in hydrostatic equilibrium, it is likely that Rhea' gravity field is partly affected by nonhydrostatic anomalies.

The Effect of Inertia on Radial Flows—Application to Hydrostatic Seals

Abstract: A theoretical study of thin fluid film flows between rotating and stationary disks is presented. Inertia terms are included using an averaged method. It is assumed that inertia effects do not influence the shape of velocity profiles. It is shown that this assumption applies in many cases encountered in fluid film lubrication. The model is validated by comparison with experimental data and previous theoretical studies. A thermoelastohydrodynamic analysis of a hydrostatic seal is performed. The substantial influence of inertia terms on leakage rate prediction is demonstrated.

A Three Dimensional Wave Activity Flux Applicable to Inertio-Gravity Waves

Abstract: A three dimensional wave activity flux applicable to non-hydrostatic inertio-gravity waves in a time mean flow in a Boussinesq fluid is derived. It is shown that the flux gives the wave-action density flux relative to the local time mean flow, and additive terms in residual circulation are equal to Stokes drift under the WKB limit. The flux is a generalization of three dimensional wave activity fluxes that have been used to analyze quasi-geostrophic wave disturbances. A flux under the log-pressure coordinate system on spherical geometry applicable to global hydrostatic inertio-gravity waves is also derived.

Mantle lateral variations and elastogravitational deformations — I. Numerical modelling

Abstract: The Earth response (deformation and gravity) to tides or to surface loads is traditionally computed assuming radial symmetry in stratified earth models, at the hydrostatic equilibrium. The present study aims at providing a new earth elastogravitational deformation model which accounts for the whole complexity of a more realistic earth. The model is based on a dynamically consistent equilibrium state which includes lateral variations in density and elastic parameters, and interface topographies. The deviation from the hydrostatic equilibrium has been taken into account as a first-order perturbation. We use a finite element method (spectral element method) and solve numerically the gravitoelasticity equations. As a validation application, we investigate the deformation of the Earth to surface loads. We first evaluate the classical loading Love numbers with a relative precision of about 0.3 per cent for PREM earth model. Then we assume an ellipsoidal homogeneous incom-pressible earth with hydrostatic pre-stresses. We investigate the impact of ellipticity on loading Love numbers analytically and numerically. We validate and discuss our numerical model. At periods greater than 1 hr, the solid earth is mainly deformed by luni-solar tides and by surface loads induced by different external fluid layers (ocean, atmosphere, continental hydrology, ice volumes). This work is devoted to the analytical and numerical development to compute the response of the Earth to such forcing. The body tides have been investigated since the 19th century. In 1862, Lord Kelvin (Sir William Thomson) made the first calculation of the elastic deformation of a homogeneous incompressible earth under the action of the tidal gravitational potential (Thomson 1862). Some years later, Love (1911) studied a compressible homogeneous earth model and showed that the tidal effects could be represented by a set of dimensionless numbers, the so-called Love numbers. Takeuchi (1950) obtained a first estimation of the Love numbers by a numerical integration of the equations using a reference earth model deduced from seismology. These results have been later extended (Smith 1974; Wahr 1981) to an ellipsoidal, rotating Earth with hydrostatic pre-stresses and a liquid core, and finally the effects of mantle anelasticity have been included (Wahr & Bergen 1986; Dehant 1987). In addition to tidal forces, mass changes in the atmosphere cause deformation and mass redistribution inside the planet. The Earth's response to such forcing involves both local and global surface motions and variations in the gravity field, which may be observed in geodetic experiments. These hydrological, atmospheric or oceanic effects on the Earth's gravity field are usually modelled for a spherical Earth with hydrostatic pre-stress (e.g. Farrell 1972; Wahr et al. 1998), generally identified to the preliminary reference earth model (PREM) developed by Dziewonski & Anderson (1981). However, the internal structure of the Earth is more complex than in a spherical non-rotating elastic isotropic (SNREI) earth model like PREM. Seismology and fluid dynamic studies show that the mantle presents heterogeneous structure induced by a thermochemical convection (Davaille 1999; Gu et al. 2001; Forte & Mitrovica 2001) and a bias from hydrostatic state. Large lateral heterogeneities have taken place on a million year timescale (Courtillot et al. 2003), like the two supposed superplumes under the Pacific and South Africa superswells, or like descending slabs. These aspects of the mantle structure are classically not taken into account in the deformation models. The elastogravitational deformations are presently observed with very high accuracy. The accuracy of superconducting gravimeter and of positioning techniques (GPS, VLBI) has seen a large improvement in the last decade. Moreover, the global gravity field will be of interest in the next 10 yr with the launch of the missions GRACE (in 2002) and GOCE (in 2007), which are dedicated to gravimetry and gradiometry 1060

Gravitational equilibrium and the mass limit for dust clouds

Abstract: We show the existence of a new class of astrophysical objects where the self-gravity of the dust is balanced by the force arising from shielded electric fields on the charged dust. The problem of equilibrium dust clouds is formulated in terms of an equation of hydrostatic force balance together with an equation of state. Because of the dust charge reduction at high dust density, the adiabatic index reduces from two to zero. This gives rise to a mass limit MAS for the maximum dust mass that can be supported against gravitational collapse by these fields. If the total mass MD of the dust in the interstellar cloud exceeds MAS, the dust collapses, while in the case MD < MAS, equilibrium may be achieved. The physics of the mass limit is similar to the Chandrasekhar's mass limit for compact objects, such as white dwarfs and neutron stars.

Comparison between discrete and continuum modeling of granular spreading

Abstract: Numerical simulation of the collapse of granular material over an horizontal plane using both continuum and discrete element approach shows that continuum models based on the Long Wave Approximation (LWA) overestimate the driving forces involved during the collapse. This effect increases when the aspect ratio of the granular mass increases. Comparison between discrete and continuum simulations makes it possible to show that the assumption of hydrostatic normal stress is essentially responsible of the limits of the LWA when the aspect ratio increases. The vertical acceleration, neglected in the vertical momentum equation, is shown to play a significant role when the aspect ratio increases. As a result, numerical simulations of real geophysical flows using thin layer models will be significantly improved if a new asymptotic analysis is performed including the effects of the vertical acceleration.

X‐ray gas in the galaxy cluster Abell 2029: conformal gravity versus dark matter

Abstract: Conformal gravity has a weak-field limit that augments the Newtonian potential -GM/R by a linear potential γc 2 R/2. Mannheim has shown that an appropriate choice of y enables a satisfying fit to the flat rotation curves of large spiral galaxies and simultaneously to the rising rotation curves of low surface brightness galaxies, without invoking dark matter. Here, we extend to larger scales the comparison of Newtonian and conformal gravity by analysis of X-ray gas in the Abell 2029 galaxy cluster. The Newtonian analysis yields a mass profile rising roughly as M α R 2 from 10 10 M ⊙ at 2kpc to 10 14 M ⊙ at 200kpc, and this can be interpreted as the profile of an extensive dark matter halo that dominates the cluster potential. In conformal gravity, the potential is non-uniform inside a spherical shell, so that both interior and exterior mass distributions must be taken into account. We derive the conformal gravity potential both inside and outside a spherical shell, enabling the evaluation of potentials for spherically symmetric mass distributions. A conformal gravity analysis of X-ray gas in Abell 2029 then yields a total mass profile that rises from 10 10 M ⊙ at 2kpc to 1.4 x 10 12 M ⊙ at 30 kpc, and then remains roughly constant out to 300 kpc. With this mass profile, conformal gravity is able to bind the X-ray gas with no need for dark matter. However, integrating the X-ray gas density profile gives a baryon mass of 10 13 M ⊙ inside 200 kpc, nearly 10 times more than what is required to hold the hot gas in hydrostatic equilibrium. This discrepancy may rule out conformal gravity unless there is a significant breakdown of hydrostatic equilibrium in the outskirts of the potential well. The required velocities, V ∼ 2000 km s -1 , may be observable via Doppler profiles in high-resolution X-ray spectroscopy. It is also possible that the mass distribution outside the cluster significantly reduces conformal gravity in the cluster outskirts. Our approximate treatment of this effect suggests that it is negligible, but a more sophisticated analysis might yield a different conclusion.

Systematics in the X-ray Cluster Mass Estimators

Abstract: We examine the systematics affecting the X-ray mass estimators applied to a set of five simulated galaxy clusters. They have been processed through the X-ray Map Simulator, X-MAS, to provide Chandra-like long exposures that are analyzed to reconstruct the gas temperature, density, and mass profiles used as input. We find that at R_2500 the mass profile obtained via a direct application of the hydrostatic equilibrium equation is consistent within 1 sigma with the actual mass; although we notice this estimator shows high statistical errors due to high level of Chandra background. Instead, the poorness of the beta-model in describing the gas density profile makes the evaluated masses to be underestimated by \sim 40 per cent with respect to the true mass, both with an isothermal and a polytropic temperature profile. We also test ways to recover the mass by adopting an analytic mass model, such as those proposed by Navarro et al. (1997) and Rasia et al. (2004), and fitting the temperature profile expected from the hydrostatic equilibrium equation to the observed one. These methods and the one of the hydrostatic equilibrium equation provide a more robust mass estimation than the ones based on the beta-model. In the present work the main limitation for a precise mass reconstruction is to ascribe to the relatively high level of the background chosen to reproduce the Chandra one. After artificially reducing it by a factor of 100, we find that the estimated mass significantly underestimates the true mass profiles. This is manly due (i) to the neglected contribution of the gas bulk motions to the total energy budget and (ii) to the bias towards lower values of the X-ray temperature measurements because of the complex thermal structure of the emitting plasma.

A unified theory of balance in the extratropics

Abstract: Many physical systems exhibit dynamics with vastly different time scales. Often the different motions interact only weakly and the slow dynamics is naturally constrained to a subspace of phase space, in the vicinity of a slow manifold. In geophysical fluid dynamics this reduction in phase space is called balance. Classically, balance is understood by way of the Rossby number R or the Froude number F; either R ≪ 1 or F ≪ 1. We examined the shallow-water equations and Boussinesq equations on an f -plane and determined a dimensionless parameter _, small values of which imply a time-scale separation. In terms of R and F, ∈= RF/√(R^2+R^2 ) We then developed a unified theory of (extratropical) balance based on _ that includes all cases of small R and/or small F. The leading-order systems are ensured to be Hamiltonian and turn out to be governed by the quasi-geostrophic potential-vorticity equation. However, the height field is not necessarily in geostrophic balance, so the leading-order dynamics are more general than in quasi-geostrophy. Thus the quasi-geostrophic potential-vorticity equation (as distinct from the quasi-geostrophic dynamics) is valid more generally than its traditional derivation would suggest. In the case of the Boussinesq equations, we have found that balanced dynamics generally implies hydrostatic balance without any assumption on the aspect ratio; only when the Froude number is not small and it is the Rossby number that guarantees a timescale separation must we impose the requirement of a small aspect ratio to ensure hydrostatic balance.

Investigation of a three-phase medium with a negative parameter of nonlinearity

Abstract: Nonlinear acoustic properties of a composite medium, consisting of hollow microspheres suspended in Castor Oil, at elevated hydrostatic pressure are experimentally investigated. The acoustic nonlinear parameter B∕A of the medium is found to be highly dependent upon hydrostatic pressure. B∕A varies from a small negative value near ambient pressure to a very large negative value (around −6000) in the vicinity of 7×104 Pascals, above ambient pressure. With a further increase in hydrostatic pressure the magnitude of B∕A decreases passing through zero and finally becomes positive. Finite amplitude wave propagation in this medium at low hydrostatic pressures is characterized by waveform steepening in the backward direction leading to rarefactive shockwaves and, at high hydrostatic pressures, by steepening in the forward direction leading to compressive shockwaves. Estimates of B∕A are obtained from both the measurement of thermodynamic properties and from waveform distortion during propagation.