Showing papers on "Hydrostatic equilibrium published in 1991"

On the Pressure Gradient Force over Steep Topography in Sigma Coordinate Ocean Models

Abstract: The error in computing the pressure gradient force near steep topography using terms following (σ) coordinates is investigated in an ocean model using the family of vertical differencing schemes proposed by Arakawa and Suarez. The truncation error is estimated by substituting known buoyancy profiles into the finite difference hydrostatic and pressure gradient terms. The error due to “hydrostatic inconsistency,” which is not simply a space truncation error, is also documented. The results show that the pressure gradient error is spread throughout the water column, and it is sensitive to the vertical resolution and to the placement of the grid points relative to the vertical structure of the buoyancy field being modeled. Removing a reference state, as suggested for the atmosphere by Gary, reduces the truncation error associated with the two lowest vertical modes by a factor of 2 to 3. As an example, the error in computing the pressure gradient using a standard 10-level primitive equation model appl...

Forced nutations of the Earth: Influence of inner core dynamics: 1. Theory

Abstract: Gravitational and pressure couplings between the solid inner core and the rest of the Earth give rise to torques through which the inner core influences the nutational motions of the Earth. In view of the very small magnitude of the moment of inertia of the inner core relative to that of the the Earth as a whole, one expects from physical considerations that inclusion of the inner core in the dynamics should lead to a new nutational normal mode with a natural frequency not too far from that of the free core nutation, and to an associated weak resonance in the amplitude of forced nutations. We present here a treatment of the nutation problem for an oceanless, elastic, spheroidally stratified Earth, with the dynamical role of the inner core explicitly included in the formulation. As a preliminary to the setting-up of dynamical equations, we devote some attention to a careful definition of a suitable coordinate system and of certain basic dynamical variables. We use the approach of Sasao et al. (1980), with their system of dynamical equations enlarged by the inclusion of two additional equations which are needed to describe the rotational motion of the inner core. An extension and sharpening of a line of reasoning employed by them enables us to derive expressions for the torques which couple the mantle and the fluid outer core to the solid inner core. Solving the enlarged system of equations, we show that a new nearly diurnal eigenfrequency does emerge; a rough estimate places it not very far from the prograde annual tidal excitation frequency, so that possible resonance effects on nutation amplitudes need careful consideration. Another eigenfrequency, attributable to a wobble of the inner core, is also found; its value is estimated to be a few times smaller than the wobble frequency that the inner core would have in the absence of couplings to the rest of the Earth. Considering an expansion, in terms of resonance contributions, of the amplitude of forced nutations normalized relative to that for a corresponding rigid Earth model, we indicate how the coefficients in the expansion are related to those in expansions of the type used by Wahr (1981b). Finally, we discuss the problem of comparing observed nutation amplitudes with those computed on the basis of Earth models generated from seismological data, with special reference to the fact that the dynamical ellipticity of the Earth, as computed from published Earth models which assume the condition of hydrostatic equilibrium, differs significantly from that determined from the precession constant. Numerical results, corrections for unmodeled effects, and comparison with observational results will be dealt with in accompanying papers.

Thermally Forced Gravity Waves in an Atmosphere at Rest

Abstract: The transient linear response of a quiescent, two-dimensional, nonrotating atmosphere to prescribed heat sources and sinks is investigated. Analytical solutions of the hydrostatic Boussinesq equations are obtained for a rigid lid and for a semi-infinite region. For the rigid lid solution, vertically trapped gravity waves propagate away from the source with a speed that depends on the Brunt–Vaisala frequency and the vertical wavenumber of the heating. The amplitude of the disturbance field in the region of the forcing approaches a constant value. Two modes are of particular interest: 1) a deep fast-moving mode which is responsible for subsidence warming through the depth of the troposphere; 2) a slower moving mode which corresponds to midlevel inflow and lower- and upper-level outflows. A solution is also obtained for a semi-infinite region. Although gravity wave energy can now propagate upward, the structure of the low-level fields still shows many similarities with the rigid lid solution. An ana...

Energy Balance in the Solar Transition Region. II. Effects of Pressure and Energy Input on Hydrostatic Models

Abstract: The radiation of energy by hydrogen lines and continua in hydrostatic energy-balance models of the transition region between the solar chromosphere and corona is studied using models which assume that mechanical or magnetic energy is dissipated in the hot corona and is then transported toward the chromosphere down the steep temperature gradient of the transition region. These models explain the average quiet sun and also the entire range of variability of the Ly-alpha lines. The relations between the downward energy flux, the pressure of the transition region, and the different hydrogen emission are described.

Model atmospheres and X-ray spectra of bursting neutron stars

Abstract: Model atmosphere equations are derived which correspond to plane-parallel nongray atmospheres of very hot neutron stars in hydrostatic and radiative equilibrium. The nonlinear transfer equation used implements an exact photon redistribution function which precisely traces even those scattering events having large photon-electron energy exchange. The numerical results thus obtained encompass tables of the surface fluxes for 20 model atmospheres whose T(eff) ranges from 6 to 30 million K. The spectra of high gravity models are virtually identical with the blackbody spectrum shifted toward higher energies, while models approaching the Eddington limit develop a large low-energy hump.

Equation of state in a strong magnetic field - Finite temperature and gradient corrections

Abstract: The equation of state for condensed matter in a strong magnetic field is constructed. The regime for which statistical models and spherical Wigner-Seitz lattice cells are valid approximations is treated. The equation of state for a free nonrelativistic homogeneous electron gas in a uniform magnetic field is examined as a function of temperature, after which this treatment is refined by incorporating Coulomb interactions in a magnetic Thomas-Fermi model which allows for finite temperature. Gradient corrections to the zero-temperature equation of state are then evaluated by constructing a magnetic Thomas-Fermi-Dirac-Weizsaecker model, these corrections having a considerable effect on the zero-pressure density for matter in strong magnetic fields. Finally, the hydrostatic equilibrium equation for the surface structure of a neutron star is integrated using the presently computed equations of state. 52 refs.

Model Generation of Spurious Gravity Waves due to Inconsistency of the Vertical and Horizontal Resolution

Abstract: The importance of the consistency between the vertical and horizontal resolution of numerical models has been suggested in recent studies In this context, consistency means that the vertical scales that are physically related to the resolvable horizontal scales are also resolved In this study, gravity waves produced in a hydrostatic primitive-equation numerical simulation of conditional symmetric instability (CSI) are shown to be produced by the inconsistency of the model resolution, where the physical relationship between the vertical and horizontal scales is determined by the slope of the narrow thermal structures (loosely termed “fronts”) produced by the CSI The detailed examination of the spurious gravity waves in the numerical simulation and height perturbations in diagnostic experiments quantify the effects of this inconsistency in the resolution It is shown that 1) spurious height perturbations of ∼1 m or less are produced, though these may be sufficient to cause significant gravity wa

A new inversion for the hydrostatic stratification of the sun

Abstract: Inversions for the spherically symmetric component of the hydrostatic stratification of the Sun are presented These employ the Backus-Gilbert optimally localized averaging procedure applied to oscillation multiplet frequencies in the range 15 – 3 mHz of modes with 4 ≤ l ≤ 140 determined by Libbrecht et al (1990) from observations carried out in 1986 We also obtain an estimate of the helium abundance in the solar convective envelope

A Mesoscale Numerical Model Using the Nonhydrostatic Pressure-based Sigma-Coordinate Equations: Model Experiments with Dry Mountain Flows

Abstract: A nonhydrostatic numerical model suitable for simulating mesoscale meteorological phenomena is developed and described here. The model is the first to exploit the nonhydrostatic equation system in σ (normalized pressure) coordinates. In addition to the commonly recognized advantages of σ-coordinate models, this model is potentially advantageous in nesting with large-scale σ-coordinate models. The equation system does not support sound waves but it presents the internal gravity waves accurately. External gravity waves are the fastest wave modes in the system that limit the integration time step. However, since short nonhydrostatic external waves are much slower than the speed of shallow-water waves and because fast hydrostatic long waves imposes less severe restriction on the time step when they are resolved by many grid points, a large time step (compared to that determined by the speed of hydrostatic shallow-water waves) can be used when horizontal grid spacing is on the order of 1 km. The syste...

A study of nonhydrostatic effects in idealized sea breeze systems

Abstract: Nonhydrostatic effects in two-dimensional mesoscale sea breeze systems are investigated by numerical simulations. It is shown that nonhydrostatic effects are directly contributed by the vertical gradients of the vertical velocity variance as well as by the vertical accelerations. It is also shown that a K-type turbulence closure is not suitable in a nonhydrostatic primitive equation model, and a higher-order closure scheme should therefore be used. Results from hydrostatic and fully-nonhydrostatic models are compared for various surface and atmospheric background conditions, such as scale and strength of surface heating, geostrophic wind, stability, surface roughness contrast, Coriolis effect, etc. It is found that for strongly developed sea breeze cases, vertical gradients of vertical velocity variance contribute most to nonhydrostatic forcing in the lower layers, and that the resultant nonhydrostatic pressure gradient acts against the hydrostatic pressure gradient, so that nonhydrostatic simulations produce weaker systems than hydrostatic ones. For weak sea breeze systems, the difference between the two models tends to be small.

Pressure drag and momentum fluxes due to the Alps. II: Representation in large‐scale atmospheric models

Abstract: A three-dimensional numerical model with interactive grid nesting is used to study the pressure drag and associated momentum fluxes due to the Alpine complex (for a specified upstream wind and static stability profile). Four basic questions are addressed which are relevant to the problems of flow description in the vicinity of mountains and the impact of orography in the context of mesoscale numerical weather prediction and general circulation models: (i) What is the dependence of basic quantities such as the pressure drag on the model's horizontal resolution? (ii) Is there a need for non-hydrostatic dynamics to account properly for the flow characteristics near steep mountains? (iii) Is there an effective role for enhanced orography such as an envelope formulation? (iv) Might a ‘gravity-wave-drag’ parametrization scheme be able to represent adequately the highest resolution fluxes and surface drag in a much coarser resolution model? A series of experiments with both hydrostatic and non-hydrostatic models at resolutions of 80, 40, 20, 10 and 5 km is analysed, and two of these experiments are repeated with an envelope orography. It is concluded that hydrostatic simulations are sufficient to describe the basic dynamics even at 5 km resolution; however, there is a strong dependence of the drag and momentum fluxes on resolution, with the drag at 80 km resolution being almost half that at the highest resolution. It is also shown that the use of an envelope orography at coarser resolutions can compensate significantly for the drag under-estimation, but is less effective in representing the vertical stress profiles or the direction of the stress vectors, which are systematically in error at the coarsest resolution. The potential role of gravity wave drag is confirmed and possible improvements considered. It is suggested that both envelope orography and gravity wave drag may make a significant contribution until resolutions are as high as 10 to 20 km.

Gasdynamic modeling of the Venus magnetotail

Abstract: A gasdynamic, convected magnetic field model of the solar wind interaction with Venus is used to model the steady state Venus magnetotail. The flow obstacle surface is approximated as a tangential discontinuity. An initial obstacle shape is defined by balancing a hydrostatic equilibrium approximation for the internal plasma pressure with an external flow pressure approximation. These approximations produce a cylindrical obstacle in the distant tail. A refined obstacle shape that attempts to balance this internal pressure with the calculated external flow pressure tapers inward toward the tail axis downstream of the terminator. The bulk plasma flow and magnetic field properties compare well with experimental observations. The model predicts central magnetotail oxygen plasma number densities of about 0.2/cu cm and temperatures on the order of 10 to the 6th K flowing tailward at speeds as low as 200 m/s.

Explosion of a rotating neutron star near the minimum mass

Abstract: A Newtonian calculation is presented for the explosion of an unstable neutron star about the minimum mass, and the amount of neutrino, gravitational, and electromagnetic radiation emitted is estimated. The dynamical evolution of the star is followed by solving the Newtonian equations or motion for a homogeneous, uniformly rotating spheroid with internal pressure and gravity. At the outset, the oblate star is in hydrostatic equilibrium at the minimum mass along an equilibrium curve of fixed angular momentum. The explosion is almost instantaneous, with an intense burst of antineutrinos signaling the onset of abrupt acceleration. Antineutrino luminosities of 10 to the 50th to 10 to the 52nd ergs/s and bulk kinetic energies of order 10 to the 49th ergs are obtained. The results indicate that the gravitational radiation (GR) energy release from the explosion is rather small at E(GR)/M(B) less than 10 to the -14th with an amplitude at maximum acceleration of less than about 5 x 10 to the -23rd sine-squared Theta for a source at a distance of 10 kpc. 19 refs.

Hydrostatic correction and pressure drop measurement in mixed convection heat transfer in a vertical tube

Abstract: In vertical, internal, mixed convection flows at high Grashof numbers and low Reynolds numbers, large radial and axial temperature profiles results in property variations which can lead to significant error in the hydrostatic correction applied to pressure drop measurementsThe improvement developed here consists of integrating and equation for the mixed mean temperature as a function of axial distance in order to obtain a better estimate for the volume-average temperature, from which a more accurate density for use in the hydrostatic correction can be determined

Measurement of Rotordynamic Coefficients for a Hydrostatic Radial Bearing

Abstract: Rockwell International, Rocketdyne Division, Canoga Park, CA 91304 Measurement of rotordynamic coefficients is presented for a pair of hydrostatic radial bearings, including direct and cross-coupled stiffness and damping. Two different hydrostatic configurations were tested: (1) an externally fed bearing 74.7 mm (2.95 in.) in diameter with a nominal direct stiffness of approximately 210 MNlm (1.2 million Ib/in.) and (2) an internally fed bearing 54.6 mm (2.15 in.) in diameter with a nominal direct stiffness of approximately 88 MNlm (0.5 million Ib/in.). Each bearing had 6 equally spaced hydrostatic pressure pockets, stationary for the externally fed bearing and rotating for the internally fed bearing. Also, both bearings had extended exit regions to provide additional damping. The top rotational speed was 22,700 rpm and the maximum axial Reynolds number was 50,000 using afreon derivative, Freon-113, as the working fluid. The test apparatus was a \"syn_ chronous rig\" as an intentionally eccentric journal was used as the sole source of excitation. Data reduction was done by performing a matrix solution to separate damping from stiffness. Results show the internally fed bearing to be 20 percent less stiff than predicted, and to have a significant amount of damping which agrees well with predictions. The internally fed bearing was found to be approximately 60 percent less stiff than predicted, and to be roughly neutral in terms of damping, as predicted.

Numerical simulations of thermal instabilities in stratified gases – II. Exploration of the parameter space

Abstract: The temporal evolution of density perturbations in an initially hydrostatic isothermal atmosphere consisting of an optically thin radiating compressible plasma is studied Numerical techniques are used to describe the nonlinear evolution of the perturbations, and the relative equilibrium between dynamic and thermal instabilities as governed by three independent control parameters are examined, namely, the initial density contrast of the perturbation, the ratio of the local buoyancy oscillation period to the local radiative cooling time, and the ratio of the perturbation radius to the local scaleheight Four orders of magnitude of initial density contrasts and ratios of buoyancy and cooling times, and one order of magnitude of the bubble dimensions are explored Well-defined oscillations were found to occur in a limited parameter range, and thermal instability to occur even within secondary condensations deriving from the bubble fragmentation

Fluid Compressibility Effects on the Dynamic Response of Hydrostatic Journal Bearings

Abstract: A theoretical analysis for the dynamic performance characteristics of laminar flow, capillar/orifice compensated hydrostatic journal bearings is presented. The analysis considers in detail the effect of fluid compressibility in the bearing recesses. At high frequency excitations beyond a break frequency, the bearing hydrostatic stiffness increases sharply and it is accompanied by a rapid decrease in direct damping. Also, the potential of pneumatic hammer instability (negative damping) at low frequencies is likely to occur in hydrostatic bearing applications handling highly compressible fluids. Useful design criteria to avoid undesirable dynamic operating conditions at low and high frequencies are determined. The effect of fluid recess compressibility is brought into perspective, and found to be of utmost importance on the entire frequency spectrum response and stability characteristics of hydrostatic/hybrid journal bearings.

Tests of random density models of terrestrial planets

Abstract: Random density models are analyzed to determine the low degree harmonics of the gravity field of a planet, and therefrom two properties: an axiality P_l , the percent of the degree variance in the zonal term referred to an axis through the maximum for degree l; and an angularity E_(ln) , the angle between the maxima for two degrees l, n. The random density distributions give solutions reasonably consistent with the axialities and angularities for the low degrees, l < 5, of Earth, Venus, and Moon, but not for Mars, which has improbably large axialities and small angularities. Hence the random density model is an unreliable predictor for the non‐hydrostatic second‐degree gravity of Mars, and thus for the moment‐of‐inertia, which is more plausibly close to 0.365MR^2.

Gravothermal instability of anisotropic self-gravitating gas spheres : singular equilibrium solution

Abstract: A re-investigation of linear perturbation theory is presented, which examines the hydrostatic re-adjustment of an isolated self-gravitating gas sphere to a redistribution of energy. Such a model describes a stellar system by the common equations of gas in hydrostatic equilibrium but with the effect of the anisotropic velocity distribution on the pressure gradient. We take as equilibrium models the singular isothermal solution with and without anisotropy. The total variation of the Boltzmann entropy resulting from a perturbation of the system caused by a redistribution of heat (i.e. rms kinetic energy of the stars) is calculated for anisotropic solutions to first order as well as to second order for the isotropic equilibrium

Cyclostrophic Balance in Surface Gravity Waves.

Abstract: The cyclostrophic balance (pressure forcevs. force centrifugal force) is shown to be satisfied for all fluid particles in surface gravity waves with sinusoidal form and circular particle orbits. Consequences of the cyclostrophic balance are 1) that the normal dispersion relation for deep water hold and 2) that the orbital radius decrease with increasing depth at the usual exponential rate, from which it follows that the wave pressure and particle speed also decrease with depth exponentially. In addition, the cyclostrophic and hydrostatic balances together predict wave breaking at the crests for amplitudes exceeding one divided by the wave number. In contrast to the traditional perturbation method, based on irrotational flow, the cyclostrophic method does not demand that the amplitude be much less than a wave length and does not require an infinite wave train.

The hydraulics problem in slope stability analysis

Abstract: Hydraulic pressures and forces are obtained on a typical slice, as used in current methods of slope stability analysis, under hydrostatic and flowing groundwater conditions It is shown that current methods do not satisfy some basic hydraulics and soil mechanics principles The effective normal stress on the lower boundary of a slice is shown to be underestimated, and the resultant hydraulic force is not accounted for adequately Key words: effective stress, hydraulic pressure, hydraulic gradient, hydraulic force, slope stability

A comparison between hydrostatic and nonhydrostatic simulations of gravity currents and their interaction with orography

Abstract: The implications of the hydrostatic assumption for an anelastic two-dimensional numerical model of a gravity current intruding into a neutrally stratified environment are studied. Particular interest is focused on situations where the gravity current encounters an orographic barrier. Considerable discrepancies between hydrostatic and nonhydrostatic calculations are found in the head region for high Froude numbers, when the resolution is high enough to resolve the head of the current. However, the propagation speed and the depth of the feeder flow are well represented in the hydrostatic model.

Unified NLTE Model Atmospheres Including Spherical Extension and Stellar Winds: EUV — Fluxes and the He II Discrepancy in Central Stars of Planetary Nebulae

Abstract: We present NLTE model atmospheres for hot stars that avoid the artificial division between hydrostatic photosphere and supersonic stellar wind envelope. The models are of spherical geometry and extended over the entire sub- and supersonic atmosphere. They need only stellar effective temperature, gravity, radius and helium abundance as free parameters and are in principle self-consistent. We have calculated a large grid of models for Central Stars of Planetary Nebulae ( CSPN ) in the effective range from 40000 K to 100000 K. The calculated energy distributions are used to investigate the longstanding problem of the ”Zanstra - discrepancy”. It is shown that the increase of EUV flux in the unified models, which is caused by the presence of the wind velocity field in the layers of He II- groundstate continuum formation leads to a much better agreement with the observations than was the case before, when plane-parallel hydrostatic models were used. However, for a significant fraction of the Central Stars investigated ( 45% ) the computed flux is still about 1 dex too small. This remaining discrepancies are attributed to additional EUV — photons generated by shocks in the radiation driven winds ( Pauldrach et al. in prep. ).

Reducing hydrostatic truncation error in a mesobeta boundary layer model

Abstract: Tiny pressure gradient forces caused by hydrostatic truncation error can overwhelm minuscule pressure gradients that drive shallow nocturnal drainage winds in a mesobeta numerical model. In seeking a method to reduce these errors, a mathematical formulation for pressure gradient force errors was derived for a single coordinate surface bounded by two pressure surfaces. A nonlinear relationship was found between the lapse rate of temperature, the thickness of the bounding pressure layer, the slope of the coordinate surface and the location of the coordinate surface within the pressure layer. The theory shows that pressure gradient force error can be reduced in the numerical model if column pressures are sums of incremental pressures over shallow layers. A series of model simulations verify the theory and show that the theory explains the only source of pressure gradient force error in the model.

A note on instabilities of a Kelvin-Helmholtz velocity profile in different approximations

Abstract: The Kelvin-Helmholtz problem deals with the stability of superposed fluids in relative motion. For a basic discontinuity in the velocity of 2U and a constant Brunt-Vaisala frequencyN it has been found that in the Boussinesq approximation small perturbations with a horizontal wavenumberk, such thatk 2>N 2/2U 2, are unstable. When the fluid is assumed to be compressible and taking into account the variation of the density in all the equations we find that there are also instabilities fork 2

Mesoscale Vertical Velocities Generated by Stress Changes in the Boundary Layer: Linear Theory

Abstract: We evaluate the mesoscale vertical velocity induced by stress changes in the surface layer as a function of the size of the rough patch in relation to environmental parameters. The nature of the flow per- turbation strongly depends on the relation between the width of the rough patch and the two natural scales of the flow, i.e. the inverse inertia wave number and the inverse of the Scorer parameter. When the width of the rough patch is comparable to the inverse inertia wave number or larger, the atmospheric perturbation is trapped, the vertical scale equals the depth of the stress surface layer, and the horizontal scale equals the Ross- by radius. When the width of the rough patch is larger than the inverse of the Scorer parameter, but smaller than the inverse inertia wave number, the atmospheric perturbation is a hydrostatic gravity wave with a verti- cal wave number equal to the Scorer parameter. When the width of the rough patch is comparable to the inverse of the Scorer parameter, the atmospheric per- turbation is a propagating lee wave with a vertical wave number equal to the Scorer parameter. When the ambient flow is strong over a small rough patch, the flow is irrotational. The same limitations, inherent to the linear gravity waves excited by the forcing in the atmosphere (e.g. mountain waves, gravity waves ini- tiated by convection, etc.), apply to the mesoscale per- turbation induced by a rough patch.

A permeameter for unsaturated soil

Abstract: A permeameter for unsaturated soil was developed by observing the way in which pore water recovers hydrostatic equilibrium It works like an hour glass that is turned upside-down everytime the state of reference (or hydrostatic equilibrium) is reached The hydraulic conductivity is deduced from the curves of evolution of pore-water pressure and from the distribution of partial density of water at hydrostatic equilibrium

Analytic Solar Structure

Abstract: In this paper an attempt is made to derive an analytic solar model by assuming a one-parameter family of density distributions. The analytic representation of the solar interior is derived for hydrostatic equilibrium and energy conservation. The mathematics involved in equating the solar luminosity to the thermonuclear energy generation near the center is illustrated in terms of special functions.