Showing papers on "Hydrostatic equilibrium published in 2008"

An Upper Gravity-Wave Absorbing Layer for NWP Applications

Abstract: Although the use of a damping layer near the top of a computational model domain has proven effective in absorbing upward-propagating gravity-wave energy in idealized simulations, this technique has been less successful in real atmospheric applications. Here, a new technique is proposed for nonhydrostatic model equations that are solved using split-explicit time-integration techniques. In this method, an implicit Rayleigh damping term is applied only to the vertical velocity, as a final adjustment at the end of each small (acoustic) time step. The adjustment is equivalent to including an implicit Rayleigh damping term in the vertical momentum equation together with an implicit vertical diffusion of w, and could be applied in this manner in other time-integration schemes. This implicit damping for the vertical velocity is unconditionally stable and remains effective even for hydrostatic gravity waves. The good absorption characteristics of this layer across a wide range of horizontal scales are confirmed through analysis of the linear wave equation and numerical mountain-wave simulations, and through simulations of an idealized squall line and of mountain waves over the Colorado Rocky Mountains.

Consistent Simulations of Substellar Atmospheres and Nonequilibrium Dust Cloud Formation

Abstract: We aim to understand cloud formation in substellar objects. We combined our nonequilibrium, stationary cloud model DRIFT (seed formation, growth, evaporation, gravitational settling, element conservation) with the general-purpose model atmosphere code PHOENIX (radiative transfer, hydrostatic equilibrium, mixing-length theory, chemical equilibrium) in order to consistently calculate cloud formation and radiative transfer with their feedback on convection and gas-phase depletion. We calculate the complete 1D model atmosphere structure and the chemical details of the cloud layers. The DRIFT-PHOENIX models enable the first stellar atmosphere simulation that is based on the actual cloud formation process. The resulting (T, p) -profiles differ considerably from the previous limiting PHOENIX cases DUSTY and COND. A tentative comparison with observations demonstrates that the determination of effective temperatures based on simple cloud models has to be applied with care. Based on our new models, we suggest a mean Teff = 1800 K for the L dwarf twin-binary system DENIS J0205–1159, which is up to 500 K hotter than suggested in the literature. We show transition spectra for gas-giant planets which form dust clouds in their atmospheres and evaluate photometric fluxes for a WASP-1 type system.

Consistent simulations of substellar atmospheres and non-equilibrium dust-cloud formation

Abstract: We aim to understand cloud formation in substellar objects. We combined the non-equilibrium, stationary cloud model of Helling, Woitke & Thi (2008; seed formation, growth, evaporation, gravitational settling, element conservation) with the general-purpose model atmosphere code PHOENIX (radiative transfer, hydrostatic equilibrium, mixing length theory, chemical equilibrium) in order to consistently calculate cloud formation and radiative transfer with their feedback on convection and gas phase depletion. We calculate the complete 1D model atmosphere structure and the chemical details of the cloud layers. The DRIFT-PHOENIX models enable the first stellar atmosphere simulation that is based on the actual cloud formation process. The resulting (T,p) profiles differ considerably from the previous limiting PHOENIX cases DUSTY and COND. A tentative comparison with observations demonstrates that the determination of effective temperatures based on simple cloud models has to be applied with care. Based on our new models, we suggest a mean Teff=1800K for the L-dwarf twin-binary system DENIS J0205-1159 which is up to 500K hotter than suggested in the literature. We show transition spectra for gas-giant planets which form dust clouds in their atmospheres and evaluate photometric fluxes for a WASP-1 type system.

Hydrodynamic planetary thermosphere model: 1. Response of the Earth's thermosphere to extreme solar EUV conditions and the significance of adiabatic cooling

Abstract: [1] It has been suggested that the exobase temperature of early terrestrial planetary atmosphere could have reached over 10,000 K. Although such high exobase temperatures should have caused the major gases at the exobase to experience fast Jeans escape, and the entire thermosphere should have experienced hydrodynamic flow, hydrostatic equilibrium was assumed to be valid in this earlier model. In this paper we develop a multicomponent hydrodynamic thermosphere model to self-consistently study the Earth's thermosphere under extreme solar EUV conditions. The model is validated against observations and other models for the present Earth's thermosphere. Simulations show that if forced in hydrostatic equilibrium and maintaining the current composition, the Earth's thermosphere could experience a fast transition to an atmospheric blowoff state when exposed to solar EUV radiation stronger than certain critical flux. When hydrodynamic flow and its associated adiabatic cooling are included, atmospheric blowoff is prevented and Earth's exobase temperature decreases with increasing solar EUV beyond the critical solar EUV flux. Simulations show that the transition of the thermosphere from the hydrostatic equilibrium regime to the hydrodynamic regime occurs when the exobase temperature reaches 7000 to 8000 K if atomic O and N dominate the upper thermosphere. The fast variations of the bulk motion velocities under different exobase temperatures suggest that the adiabatic cooling effect could have kept the exobase temperature lower than ∼1000 K if light gases such as atomic hydrogen were the dominant species in the Earth's thermosphere. We propose that hydrodynamic flow and associated adiabatic cooling should exist in the thermospheres of a broad range of early and/or close-in terrestrial type planets and that the adiabatic cooling effect must be included in the energy balance in order to correctly estimate their thermospheric structures and their evolutionary paths.

Assessment of the non-hydrostatic effect on the upper atmosphere using a general circulation model (GCM)

Abstract: [1] Under hydrostatic equilibrium, a typical assumption used in global thermosphere ionosphere models, the pressure gradient in the vertical direction is exactly balanced by the gravity force. Using the non-hydrostatic Global Ionosphere Thermosphere Model (GITM), which solves the complete vertical momentum equation, the primary characteristics of non-hydrostatic effects on the upper atmosphere are investigated. Our results show that after a sudden intense enhancement of high-latitude Joule heating, the vertical pressure gradient force can locally be 25% larger than the gravity force, resulting in a significant disturbance away from hydrostatic equilibrium. This disturbance is transported from the lower altitude source region to high altitudes through an acoustic wave, which has beensimulated inaglobal circulation model forthefirst time. Due to the conservation of perturbation energy, the magnitude of the vertical wind perturbation increases with altitude andreaches 150(250) m/sat300(430) kmduringthe disturbance. The upward neutral wind lifts the atmosphere and raises the neutral density at high altitudes by more than 100%. These large vertical winds are not typically reproduced by hydrostatic models of the thermosphere and ionosphere. Our results give an explanation of the cause of such strong vertical winds reported in many observations. Citation: Deng, Y., A. D. Richmond, A. J. Ridley, and H.-L. Liu (2008), Assessment of the non-hydrostatic effect on the upper atmosphere using a general circulation model (GCM), Geophys. Res. Lett., 35, L01104, doi:10.1029/2007GL032182.

Pressure Relations and Vertical Equilibrium in the Turbulent, Multiphase ISM

Abstract: We use numerical simulations of turbulent, multiphase, self-gravitating gas orbiting in model disk galaxies to study the relationships among pressure, the vertical gas distribution, and the ratio of dense to diffuse gas. We show that the disk height and mean midplane pressure are consistent with effective hydrostatic equilibrium, provided that the turbulent vertical velocity dispersion and gas self-gravity are included. Mass-weighted pressures are an order of magnitude higher than the midplane pressure because self-gravity concentrates gas and increases the pressure in clouds. We also investigate the ratio Rmol=M(H2)/M(HI) for our simulations. Blitz and Rosolowsky (2006) showed that Rmol is proportional to the estimated midplane pressure. For model series in which the epicyclic frequency, kappa, and gas surface density, Sigma, are proportional, we recover the empirical relation. For other model series in which kappa and Sigma are independent, the midplane pressure and Rmol are not well correlated. We conclude that the molecular fraction -- and star formation rate -- of a galactic disk inherently depends on its rotational state, not just the local values of Sigma and the stellar density rho*. The empirical correlation between Rmol and midplane pressure implies that the "environmental parameters" kappa, Sigma, and rho* are interdependent in real galaxies, presumably as a consequence of evolution toward states with Toomre Q near unity. We note that Rmol in static models far exceeds both the values in our turbulent simulations and observed values, implying that turbulence is crucial to obtaining a realistic molecular fraction in the ISM.

Planet Shadows in Protoplanetary Disks. I. Temperature Perturbations

Abstract: Planets embedded in optically thick passive accretion disks are expected to produce perturbations in the density and temperature structure of the disk. We calculate the magnitudes of these perturbations for a range of planet masses and distances. The model predicts the formation of a shadow at the position of the planet paired with a brightening just beyond the shadow. We improve on previous work on the subject by self-consistently calculating the temperature and density structures under the assumption of hydrostatic equilibrium and taking the full three-dimensional shape of the disk into account rather than assuming a plane-parallel disk. While the excursion in temperatures is less than in previous models, the spatial size of the perturbation is larger. We demonstrate that a self-consistent calculation of the density and temperature structure of the disk has a large effect on the disk model. In addition, the temperature structure in the disk is highly sensitive to the angle of incidence of stellar irradiation at the surface, so accurately calculating the shape of the disk surface is crucial for modeling the thermal structure of the disk.

Radiation Pressure-Supported Active Galactic Nucleus Tori with Hard X-Ray and Stellar Heating

Abstract: The dynamics and structure of toroidal obscuration around active galactic nuclei remain uncertain and controversial. In this paper we extend earlier work on the dynamical role of infrared radiation pressure by adding the effects of two kinds of distributed heating: Compton heating due to hard X-rays from the nucleus and local starlight heating. We find numerical solutions to the axisymmetric hydrostatic equilibrium, energy balance, and photon diffusion equations including these effects. Within the regime of typical parameters, the two different sources of additional heating have very similar effects: the density profile within the torus becomes shallower both radially and vertically, but for plausible heating rates, there is only minor change (relative to the source-free case) in the distribution of column density with solid angle. The most interesting consequence of distributed heating is that it selects out a relatively narrow range of parameters permitting an equilibrium, particularly -->(L/LE)/τT. We discuss the implications of both the narrowness of the permitted range and its approximate coincidence with the range inferred from observations.

Radiation Pressure Supported AGN Tori with Hard X-Ray and Stellar Heating

Abstract: The dynamics and structure of toroidal obscuration around AGN remain uncertain and controversial. In this paper we extend earlier work on the dynamical role of infrared radiation pressure by adding the effects of two kinds of distributed heating: Compton-heating due to hard X-rays from the nucleus and local starlight heating. We find numerical solutions to the axisymmetric hydrostatic equilibrium, energy balance, and photon diffusion equations including these effects. Within the regime of typical parameters, the two different sources of additional heating have very similar effects: the density profile within the torus becomes shallower both radially and vertically, but for plausible heating rates, there is only minor change (relative to the source-free case) in the distribution of column density with solid angle. The most interesting consequence of distributed heating is that it selects out a relatively narrow range of parameters permitting an equilibrium, particularly $(L/L_E)/\tau_T$. We discuss the implications of both the narrowness of the permitted range and its approximate coincidence with the range inferred from observations.

A non-hydrostatic Rhea

Abstract: [1] Radiometric data obtained during Cassini's close flyby of Rhea, on 26 November 2005, has been subject to several published analyses aiming to determine the satellite's mass and quadrupole gravity moments. Combining aspects of two of these analyses we present our best, unbiased estimates of the gravity field parameters and point out how the constraint of hydrostatic equilibrium adopted by previous analysts affects the results. We present solutions based on a broad range of geophysical assumptions, such as the presence of degree 3 and 4 gravity field constrained at different levels. The result is a balanced approach which describes our current knowledge of Rhea's gravity field. In the case of a gravity field limited to second degree harmonics the most reliable estimates are GM = 153.9398 ± 0.0008 km3 s−2, 106J2 = 931.0 ± 12.0, 106C22 = 237.2 ± 4.5, and 106S22 = 3.8 ± 3.8.

Nonlinear dynamics of hydrostatic internal gravity waves

Abstract: Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.

Simulation of Hydrostatical Equilibrium of Ferrofluid Subject to Magneto-static Field

Abstract: In this paper, a numerical scheme for obtaining ferrofluid shape formation is presented using the revisited Kelvin's force formula. The formula adopts the newly appreciated field intensity from the original Kelvin's concept and the virtual air gap scheme. For the hydrostatic equilibrium subject to magneto-static field, both electromagnetic and gravitational body forces should be considered. The free surface profile of ferrofluid can be obtained through numerical iterations based on the fact that the pressures on the free surface should be the same. Test models are presented as validations of the proposed scheme.

Hydrostatic compression model for sandy soils

Abstract: Based on elastic considerations, a new four parameter model of the hydrostatic compression curves of sandy soils is presented. Using a unique set of parameters, this model is capable of accurately representing the infinite family of compression curves that result when soil is subjected to stresses below particle breakage. The parameters are readily estimated from two hydrostatic tests conducted at different initial densities during loading and unloading stages. The resulting equations are able to predict the effect of progressive stiffness increase and compressibility reduction that occur as void ratio decreases during both loading and unloading conditions. The same normalized stress scale employed for the hydrostatic compression curves can also be used to describe the steady state line, which leads to a unique framework for representing the stress–void ratio curves of sandy soils. A comprehensive series of simulations on 13 different types of sandy soils in both loose and dense states was performed under...

Three-Dimensional Non-Hydrostatic Model for Free-Surface Flows With Unstructured Grid

Abstract: The three-dimensional Navier-Stokes equations were solved with the fractional step method where the hydrostatic pressure component was determined first, while the non-hydrostatic component of the pressure was computed from the pressure Poisson equation in which the coefficient matrix is positive definite and symmetric. The eddy viscosity was calculated from the efficient k-ɛ turbulence model. The resulting model is computationally efficient and unrestricted to the CFL condition. Computations with and without hydrostatic approximation were compared for the same cases to test the validity of the conventional hydrostatic pressure assumption. The model was verified against analytical solutions and experimental data, with excellent agreement.

Symmetric Stability of Compressible Zonal Flows on a Generalized Equatorial β Plane

Abstract: Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Co...

An Evolving Entropy Floor in the Intracluster Gas

Abstract: Nongravitational processes, such as feedback from galaxies and their active nuclei, are believed to have injected excess entropy into the intracluster gas, and therefore to have modified the density profiles in galaxy clusters during their formation. Here we study a simple model for this so-called preheating scenario, and ask (1) whether it can simultaneously explain both global X-ray scaling relations and number counts of galaxy clusters, and (2) whether the amount of entropy required evolves with redshift. We adopt a baseline entropy profile that fits recent hydrodynamic simulations, modify the hydrostatic equilibrium condition for the gas by including ≈20% nonthermal pressure support, and add an entropy floor K0 that is allowed to vary with redshift. We find that the observed luminosity-temperature (L − T) relations of low-redshift ( z = 0.05) HIFLUGCS clusters and high-redshift ( z = 0.80) WARPS clusters are best simultaneously reproduced with an entropy floor that evolves from ≈200 h−1/3 keV cm 2 at z ≈ 0.8 to 300 h−1/3 keV cm 2 at z < 0.05. This evolution may take place predominantly at low redshift (z 0.2). If we restrict our analysis to the subset of bright (kT 3 keV) clusters, we find that the evolving entropy floor can mimic a self-similar evolution in the L − T scaling relation. This degeneracy with self-similar evolution is, however, lifted when 0.5 keV kT 3 keV clusters are included. Using the cosmological parameters from the WMAP 3 yr data, but treating σ8 as a free parameter, our model can reproduce the number counts of the X-ray galaxy clusters in the 158 deg2 ROSAT PSPC survey, with a best-fit value of σ8 = 0.80 ± 0.05.

Nonlinear Dynamic Behavior of a Rigid Rotor Supported by Hydrostatic Squeeze Film Dampers

Abstract: This research project aims to study the nonlinear dynamic behavior of a rigid rotor supported by hydrostatic squeeze film dampers (HSFDs). The investigated HSFD consists of four hydrostatic bearing flat pads fed by capillary restrictors. A nonlinear hydrostatic squeeze film damper model is developed, and the results are compared with those obtained using a linear approach. The effect of unbalance eccentricity on the vibration response and the transmitted force of the HSFD are investigated using the linear and nonlinear models. The results show good agreement between the linear and nonlinear methods when the unbalance force is small. However, as the unbalance forces become larger, the results obtained using the linear models cease to be representative of the real behavior of rotor dynamics and a nonlinear approach must be conducted. The effects of supply pressure, viscosity, pressure ratio, and rotational speed on the response and the force transmitted to the HSFD are investigated using a nonlinear approach.

A generalized thermal wind equation and some non‐separable exact solutions of the flow equations for three‐dimensional spherical atmospheres

Abstract: Steady axisymmetric flows in geostrophic and cyclostrophic balance under gravity are considered for the three-dimensional Euler equations (with the spherical geopotential approximation but without the shallow-atmosphere and ‘traditional’ approximations). The key to analytical specification of these flows is a compatibility condition involving the zonal flow and temperature fields. This condition is a generalization of the thermal wind equation for balanced zonal flow governed by the hydrostatic primitive equations. Two examples are presented in which the temperature field is specified as non-separable two-parameter functions of latitude and height, and corresponding zonal flows are derived analytically. Such flows extend the class of known analytical solutions of the governing equations, and their usefulness in numerical model development and testing is the focus of this study. © Crown Copyright 2008. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd.

On the Expansion of Quasi-Static, Twisted Coronal Loops in Uniform Gravity

Abstract: Coronal loop emission profiles are often of remarkably constant width along their entire lengths, contradicting expectations based on model coronal magnetic field strengths decreasing with height. Meanwhile P. Bellan has produced a theoretical model in which an initially empty, twisted force-free loop, on being filled with plasma via upflow at each footpoint, in the absence of significant gravitational effects, forms a narrow, filamentary loop of constant cross section. In this paper, we focus on equilibrium states that include stratification by uniform gravity while retaining the effects of magnetic field twist. Comparing these with related force-free equilibria, it is found that injection of low-? plasma under coronal conditions is not likely to change the shape of a loop significantly. These linear equilibria apply to the interiors and boundaries of loops only, with external influences modeled by boundary total pressures. The effects of total pressure balance with surroundings and of gravitational stratification are to inhibit the pinching of a loop to a constant cross section. Only if the plasma ? were high enough for the plasma to reconfigure the external field and the hydrostatic scale height much greater than the loop size could the final state have nearly constant cross section. We do not expect this to occur in the corona.

Start and stop of an avalanche in a granular medium subjected to an inner water flow.

Abstract: We report experimental results on the maximum angle of stability, i.e., the so-called avalanche angle, of a granular medium subjected to an inner water flow controlled by a constant pressure drop. A unique avalanche threshold is derived by two alternative theoretical developments, namely a continuum and a discrete approach, and is successfully confronted to many measurements in a large experimental range. A qualitative analysis of the instability triggering reveals different dynamical behaviors depending on whether the water flow is downward or upward in the granular layer, namely stabilizing versus destabilizing regime. Contrary to the purely hydrostatic situation, the free surface following an avalanche departs from a linear shape because the dynamical pressure gradient is no longer constant in the medium. A simple model is proposed that can satisfactorily predict the postavalanche height profile as well as its subsequent evolution for higher inclinations.

General relationships between pressure, weight and mass of a hydrostatic fluid

Abstract: In curved geometries the hydrostatic pressure in a fluid does not equal the weight per unit area of the fluid above it. General weight–pressure and mass–pressure relationships for hydrostatic fluids in any geometry are derived. As an example of the mass–pressure relationship, we find a geometric reduction in surface pressure as large as 5 mbar on Earth and 39 mbar on Titan. We also present a thermodynamic interpretation of the geometric correction which, as a corollary, provides an independent proof of the hydrostatic relationship for general geometries.

Model simulation of the global circulation in the middle atmosphere for January conditions

Abstract: . A mathematical model of the global neutral wind system of the Earth's atmosphere, developed earlier in the Polar Geophysical Institute (PGI), is utilized to simulate the large-scale global circulation of the middle atmosphere for January conditions. The utilized model enables to calculate not only the horizontal components but also the vertical component of the neutral wind velocity by means of a numerical solution of a generalized Navier-Stokes equation for compressible gas, so the hydrostatic equation is not applied. Global distributions of the horizontal and vertical wind, calculated for January conditions, are compared with simulation results, obtained earlier for conditions corresponding to summer in the northern hemisphere. It was found that the global distributions of the neutral wind, calculated both for winter and for summer periods in the northern hemisphere, in particular, the large-scale circumpolar vortices, are consistent with the planetary circulation of the atmosphere, obtained from observations.