Hydrostatic equilibrium

About: Hydrostatic equilibrium is a research topic. Over the lifetime, 2451 publications have been published within this topic receiving 62172 citations.

Estimation of Formation Pressures from Log-Derived Shale Properties(*)

Abstract: Sedimentary rocks during burial maintain hydrostatic fluid pressure within their pore space in the fluids within the sediment are allowed to escape as the sediment compacts. If the fluid is not permitted to escape, compaction is retarded, and the fluid pressure rises (the sediment becomes overpressured) and ultimately approaches the pressure exerted by the overlying rocks and contained fluids. The actual fluid pressure existing in a given permeable formation can be determined by standard pressure bomb measurements. The determination of the fluid pressure in shales, with their low permeability, has previously been difficult or impossible. The fluid pressure within the pore space of shales can be determined by using data obtained from both acoustic and resistivity logs. The method involves establishing relationships between the common logarithm of shale transit time or shale resistivity and depth for hydrostatic-pressured formations. On a plot of transitime versus depth, a linear relationship is generally observed, whereas on a plot of resistivity versus depth, a nonlinear trend exists. Divergence of observed transit time or resistivity values from those obtained from established normal compaction trends under hydrostatic pressure conditions is a measure of the pore fluid pressure in the shale and thus in adjacent isolated permeable formations. This relationship has been empirically established with actual pressure measurements in adjacent permeable formations. The use of these data and this method permits the interpretation of fluid pressure from acoustic and resistivity measurements with an accuracy of approximately 0.04 psi per foot, or about 400 psi at 10,000 feet. The standard deviation for the resistivity method is 0.022 psi per foot and for the acoustic method 0.020 psi per foot. Knowledge of the first occurrence of overpressures, and indeed of the precise pressure-depth relationship in a geologic province, enables improvements in drilling techniques, casing programs, completion methods, and reservoir evaluations.

Gravitational collapse of an isothermal sphere

Abstract: We investigate the spherical gravitational collapse of isothermal spheres using numerical hydrodynamics. The initial configuration is close to hydrostatic equilibrium. In the initial density profile has a finite core radius (i.e., it is not singular), supersonic velocities develop during the initial collapse. At the time of central core formation, when the central density diverges, the central inflow velocity approaches −3.3 times the sound speed and the central density approaches an r −2 profile. These conditions are similar to those found in the self-similar solution of Larson and Penston, but occur only at the center and not at all radii as in the self-similar solution at core formation

An asymptotic theory for the interaction of waves and currents in coastal waters

Abstract: A multi-scale asymptotic theory is derived for the evolution and interaction of currents and surface gravity waves in water of finite depth, under conditions typical of coastal shelf waters outside the surf zone. The theory provides a practical and useful model with which wave–current coupling may be explored without the necessity of resolving features of the flow on space and time scales of the primary gravity-wave oscillations. The essential nature of the dynamical interaction is currents modulating the slowly evolving phase of the wave field and waves providing both phase-averaged forcing of long infra-gravity waves and wave-averaged vortex and Bernoulli-head forces and hydrostatic static set-up for the low-frequency current and sea-level evolution equations. Analogous relations are derived for wave-averaged material tracers and density stratification that include advection by horizontal Stokes drift and by a vertical Stokes pseudo-velocity that is the incompressible companion to the horizontal Stokes velocity. Illustrative solutions are analysed for the special case of depth-independent currents and tracers associated with an incident surface wave field and a vortex with O(1) Rossby number above continental shelf topography.

A dynamical model for the distribution of dark matter and gas in galaxy clusters

Abstract: Using the results of an extended set of high-resolution non-radiative hydrodynamic simulations of galaxy clusters, we obtain simple analytic formulae for the dark matter and hot gas distribution, in the spherical approximation. Starting from the dark matter phase-space radial density distribution, we derive fits for the dark matter density, velocity dispersion and velocity anisotropy. We use these models to test the dynamical equilibrium hypothesis through the Jeans equation: we find that this is satisfied to good accuracy by our simulated clusters inside their virial radii. This result also shows that our fits constitute a self-consistent dynamical model for these systems. We then extend our analysis to the hot gas component, obtaining analytic fits for the gas density, temperature and velocity structure, with no further hypothesis on the gas dynamical status or state equation. Gas and dark matter show similar density profiles down to ≈0.06Rv (with Rv the virial radius), while at smaller radii the gas flattens, producing a central core. Gas temperatures are almost isothermal out to roughly 0.2 Rv, then steeply decrease, reaching at the virial radius a value almost a factor of 2 lower. We find that the gas is not at rest inside Rv: velocity dispersions are increasing functions of the radius, motions are isotropic to slightly tangential, and contribute non-negligibly to the total pressure support. We test this model using a generalization of the hydrostatic equilibrium equation, where the gas motion is properly taken into account. Again we find that the fits provide an accurate description of the system: the hot gas is in equilibrium and is a good tracer of the overall cluster potential if all terms (density, temperature and velocity) are taken into account, while simpler assumptions cause systematic mass underestimates. In particular, we find that using the so-called β-model underestimates the true cluster mass by up to 50 per cent at large radii. We also find that, if gas velocities are neglected, then a simple isothermal model fares better at large radii than a non-isothermal one. The shape of the gas density profile at small radii is at least partially explained by the gas expansion caused by energy transfer from dark matter during the collapse. In fact, when gas bulk energy is also considered, gas and dark matter are in energy equipartition in the final system at radii r > 0.1Rv, while at smaller radii the gas is hotter than the dark matter. This energy imbalance is also probably the reason of the further global halo compression compared with a pure collisionless collapse, which we point out by comparing the dark matter and total density profiles of our hydro-simulated clusters with a set of identical – but pure N-body – ones. The compression has the effect of raising the mean concentration by an amount of roughly 10 per cent.

Modeling of Coronal EUV Loops Observed with TRACE. I. Hydrostatic Solutions with Nonuniform Heating

Abstract: Recent observations of coronal loops in EUV wavelengths with the Transition Region and Coronal Explorer (TRACE) and the Extreme-Ultraviolet Imaging Telescope (EIT) on the Solar and Heliospheric Observatory (SOHO) demonstrated three new results that cannot be explained by most of the existing loop models: (1) EUV loops are near-isothermal along their coronal segments, (2) they show an overpressure or overdensity compared with the requirements of steady state loops with uniform heating, and (3) the brightest EUV loops exhibit extended scale heights up to 4 times the hydrostatic scale height. These observations cannot be reconciled with the classical RTV (Rosner, Tucker, & Vaiana) model, they do not support models with uniform heating, and they even partially violate the requirements of hydrostatic equilibrium. In this study we are fitting for the first time steady state solutions of the hydrodynamic equations to observed intensity profiles, permitting a detailed consistency test of the observed temperature T(s) and density profiles ne(s) with steady state models, which was not possible in previous studies based on scaling laws. We calculate some 500 hydrostatic solutions, which cover a large parameter space of loop lengths (L ? 4-300 Mm), of nonuniform heating functions (with heating scale heights in the range of ?H ? 1-300 Mm), approaching also the limit of uniform heating (?H L). The parameter space can be subdivided into three regimes, which contain (1) solutions of stably stratified loops, (2) solutions of unstably stratified loops (in the case of short heating scale heights, ?H,Mm ? ), and (3) a regime in which we find no numerical solutions (when ?H,Mm ). Fitting the hydrostatic solutions to 41 EUV loops observed with TRACE (selected by the criterion of detectability over their entire length), we find that only 30% of the loops are consistent with hydrostatic steady state solutions. None of the observed EUV loops is consistent with a uniform heating function while in quasi-steady state. Those loops compatible with a steady state are found to be heated near the footpoints, with a heating scale height of ?H = 12 ? 5 Mm, covering a fraction ?H/L = 0.2 ? 0.1 of the loop length. These results support coronal heating mechanisms operating in or near the chromosphere and transition region.