Hydrostatic Equilibrium Research Articles

Stability of hydrostatic equilibrium of the 2D Boussinesq-MHD equations without magnetic diffusion in two kinds of periodic domains

Stability of hydrostatic equilibrium of the 2D Boussinesq-MHD equations without magnetic diffusion in two kinds of periodic domains

Using Icepack to reproduce ice mass balance buoy observations in landfast ice: improvements from the mushy-layer thermodynamics

Abstract. Icepack (v1.1.0) – the column thermodynamics model of the Community Ice CodE (CICE) version 6 – is used to assess how changing the thermodynamics from the Bitz and Lipscomb (1999) physics (hereafter BL99) to the mushy-layer physics impacts the model performance in reproducing in situ landfast ice observations from two ice mass balance (IMB) buoys co-deployed in the landfast ice close to Nain (Labrador) in February 2017. To this end, a new automated surface retrieval algorithm is used to determine the in situ ice thickness, snow depth, basal ice congelation and snow-ice formation from the measured vertical temperature profiles. Icepack simulations are run to reproduce these observations using each thermodynamics scheme, with a particular interest in how the different physics influence the representation of snow-ice formation and ice congelation. Results show that the BL99 parameterization represents well the ice congelation but underrepresents the snow-ice contribution to the ice mass balance. In particular, defining snow-ice formation based on the hydrostatic balance alone does not reproduce the negative freeboards observed for several days in the IMB observations, resulting in an earlier snow-flooding onset, a positive ice thickness bias and reduced snow depth variations. We find that the mushy-layer thermodynamics with default parameters significantly degrades the model performance, overestimating both the congelation growth and snow-ice formation. The simulated thermodynamics response to flooding, however, better represents the observations, and the best results are obtained when allowing for negative freeboards in the mushy-layer physics. We find that the mushy-layer thermodynamics produces a larger variability in congelation rates at the ice bottom interface, alternating between periods of exceedingly fast growth and periods of unrealistic basal melt. This pattern is related to persistent brine dilution in the lowest ice layer by the congelation and brine drainage parameterizations. We also show that the mushy-layer congelation parameterization produces significant frazil formation, which is not expected in a landfast ice context. This behavior is attributed to the congelation parameterization not fully accounting for the conductive heat flux imbalance at the ice–ocean boundary. We propose a modification of the mushy-layer congelation scheme that largely reduces the frazil formation and allows for better tuning of the congelation rates to match the observations. Our results demonstrate that the mushy-layer physics and its parameters can be tuned to closely match the in situ observations, although more observations are needed to better constrain them.

A multi-decadal time series of upper stratospheric temperature profiles from Odin-OSIRIS limb-scattered spectra

Abstract. A new upper stratospheric (35–60 km) temperature data product has been produced using Optical Spectrograph and InfraRed Imager System (OSIRIS) limb-scattered spectra that now spans over 22 years. Temperature is calculated by first estimating the Rayleigh scattering signal and then integrating hydrostatic balance combined with the ideal gas law. Uncertainties are estimated to be 1–5 K, with a vertical resolution of 3–4 km. Correlative comparisons with the Atmospheric Chemistry Experiment Fourier Transform Spectrometer (ACE-FTS) and the Microwave Limb Sounder on Aura (MLS) are consistent with these uncertainty estimates and generally have no regions of statistically significant drift. The data product has been publicly released as part of the nominal OSIRIS v7.3 processing.

CONGRuENTS (COsmic-ray, Neutrino, Gamma-ray and Radio Non-Thermal Spectra) – II. Population-level correlations between galactic infrared, radio, and γ-ray emission

ABSTRACT Galaxies obey a number of empirical correlations between their radio, γ-ray, and infrared emission, but the physical origins of these correlations remain uncertain. Here, we use the CONGRuENTS model for broad-band non-thermal emission from star-forming galaxies, which self-consistently calculates energy-dependent transport and non-thermal emission from cosmic ray hadrons and leptons, to predict radio and γ-ray emission for a synthetic galaxy population with properties drawn from a large deep-field survey. We show that our synthetic galaxies reproduce observed relations such as the far infrared (FIR)–radio correlation, the FIR–γ correlation, and the distribution of radio spectral indices, and we use the model to explain the physical origins of these relations. Our results show that the FIR–radio correlation arises because the amount of cosmic ray electron power ultimately radiated as synchrotron emission varies only weakly with galaxy star formation rate as a result of the constraints imposed on gas properties by hydrostatic balance and turbulent dynamo action; the same physics dictates the extent of proton calorimetry in different galaxies, and thus sets the FIR–γ–ray correlation. We further show that galactic radio spectral indices result primarily from competition between thermal free–free emission and energy-dependent loss of cosmic ray electrons to bremsstrahlung and escape into galactic haloes, with shaping of the spectrum by inverse Compton, synchrotron, and ionization processes typically playing a subdominant role. In addition to explaining existing observations, we use our analysis to predict a heretofore unseen correlation between the curvature of galaxies’ radio spectra and their pion-driven γ-ray emission, a prediction that will be testable with upcoming facilities.

Stability of a realistic astrophysical pulsar and its mass–radius relation in higher-order curvature gravity

The objective of this research is to explore compact celestial objects while considering the framework of an extended gravitational theory known as R+f(G) gravity. The notations R and G denote the Ricci scalar and the Gauss–Bonnet invariant, respectively. Radio pulsars, which are neutron stars with masses greater than 1.8 times that of the Sun (M⊙), provide exceptional opportunities for delving into fundamental physics in extraordinary environments unparalleled in the observable universe and surpassing the capabilities of experiments conducted on Earth. Through the utilization of both the linear and quadratic expressions of the function f(G)=α1G2, where α1 (with dimensional units of [length6]) are incorporated, we have achieved an accurate analytical solution for anisotropic perfect-fluid spheres in a state of hydrostatic equilibrium. By integrating the dimensional parameters α1 and the compactness factor, defined as C=2GMRc2, we showcase our capacity to encompass and depict all physical characteristics within the stellar structure. We illustrate that the model is capable of producing a stable arrangement for the pulsar PSR J0740+6620, encompassing its geometric and physical properties. We illustrate that by utilizing the quadratic form of G in the R+f(G) framework, the Krori-Barua ansatz establishes semi-analytical relationships between radial pressure (pr), tangential pressure (pt), and density (ρ). Remarkably, in the context of the positive/negative α1 quadratic R+f(G) gravity framework, the maximum compactness is inherently restricted to values fail below the Buchdahl limit when the surface density exceeds 4×1014 g/cm3. This differs from General Relativity (GR) in which the compactness is unlikely to approach the black hole limit. The model predicts a core density that is multiple times greater than the saturation nuclear density, represented as ρnuc=4×1014 g/cm3, and a surface density ρs that surpasses the nuclear saturation density. We offer the mass–radius diagram associated with the boundary density we have determined, and this diagram has been shown to be consistent with other observational discoveries.

Compact stars in Rastall gravity: hydrostatic equilibrium and radial pulsations

Compact stars in Rastall gravity: hydrostatic equilibrium and radial pulsations

Finch–Skea Stellar structures obeying Karmarkar condition in modified [formula omitted] gravity

The current research addresses the composition and modeling of compact dense astrophysical objects in a renowned modified gravity namely f(G) theory, where G denotes the Gauss Bonnet term. In this respect, a family of anisotropic spherically symmetric relativistic solutions of compact objects in hydrostatic equilibrium are obtained. To solve the corresponding modified field equations, we employ the well-famed Karmarkar condition along with the Finch–Skea ansatz for one of the metric potentials. By taking the outer area as Schwarzschild metric, we expand the boundary conditions for spacetime continuity. The unknown constants, appearing in the solutions, are then assessed by using data (mass and radius) of some realistic compact stars namely SMCX-1, 4U 1728-34, 4U 1538-52, EXO 1785-248, SAXJ 1808.4-3658. Furthermore, a graphical analysis is performed for examining the physical behavior of these compact star models. We discuss several physical properties of anisotropic compact stellar structures and explore viability as well as stability of the proposed models. It is concluded that features of established models are in good agreement with the reality of compact stellar structures in the background of f(G) theory.

Variable-Dependent and Selective Multivariate Localization for Ensemble–Variational Data Assimilation in the Tropics

Abstract Two aspects of ensemble localization for data assimilation are explored using the simplified nonhydrostatic ABC model in a tropical setting. The first aspect (i) is the ability to prescribe different localization length scales for different variables (variable-dependent localization). The second aspect (ii) is the ability to control (i.e., to knock out by localization) multivariate error covariances (selective multivariate localization). These aspects are explored in order to shed light on the cross-covariances that are important in the tropics and to help determine the most appropriate localization configuration for a tropical ensemble–variational (EnVar) data assimilation system. Two localization schemes are implemented within the EnVar framework to achieve (i) and (ii). One is called the isolated variable-dependent localization (IVDL) scheme and the other is called the symmetric variable-dependent localization (SVDL) scheme. Multicycle observation system simulation experiments are conducted using IVDL or SVDL mainly with a 100-member ensemble, although other ensemble sizes are studied (between 10 and 1000 members). The results reveal that selective multivariate localization can reduce the cycle-averaged root-mean-square error (RMSE) in the experiments when cross-covariances associated with hydrostatic balance are retained and when zonal wind/mass error cross-covariances are knocked out. When variable-dependent horizontal and vertical localization are incrementally introduced, the cycle-averaged RMSE is further reduced. Overall, the best performing experiment using both variable-dependent and selective multivariate localization leads to a 3%–4% reduction in cycle-averaged RMSE compared to the traditional EnVar experiment. These results may inform the possible improvements to existing tropical numerical weather prediction systems that use EnVar data assimilation.

The XXL Survey

Context. The thermodynamical properties of the intracluster medium (ICM) are driven by scale-free gravitational collapse, but they also reflect the rich astrophysical processes at play in galaxy clusters. At low masses (∼1014 M⊙) and high redshift (z ≳ 1), these properties remain poorly constrained, observationally speaking, due to the difficulty in obtaining resolved and sensitive data. Aims. We aim to investigate the inner structure of the ICM as seen through the Sunyaev–Zel’dovich (SZ) effect in this regime of mass and redshift. We focused on the thermal pressure profile and the scaling relation between SZ flux and mass, namely the YSZ − M scaling relation. Methods. The three galaxy clusters XLSSC 072 (z = 1.002), XLSSC 100 (z = 0.915), and XLSSC 102 (z = 0.969), with M500 ∼ 2 × 1014 M⊙, were selected from the XXL X-ray survey and observed with the NIKA2 millimeter camera to image their SZ signal. XMM-Newton X-ray data were used as a complement to the NIKA2 data to derive masses based on the YX − M relation and the hydrostatic equilibrium. Results. The SZ images of the three clusters, along with the X-ray and optical data, indicate dynamical activity related to merging events. The pressure profile is consistent with that expected for morphologically disturbed systems, with a relatively flat core and a shallow outer slope. Despite significant disturbances in the ICM, the three high-redshift low-mass clusters follow the YSZ − M relation expected from standard evolution remarkably well. Conclusions. These results indicate that the dominant physics that drives cluster evolution is already in place by z ∼ 1, at least for systems with masses above M500 ∼ 1014 M⊙.

The stability of Boussinesq equations with partial dissipation around the hydrostatic balance

The stability of Boussinesq equations with partial dissipation around the hydrostatic balance

Viscosity Flow Curves of Agar and the Bounded Ripening Growth Model of the Gelation Onset.

The gelation kinetics of agar aqueous solutions were studied by means of the viscosity flow curves using a coaxial Couette cylinder viscometer. The viscosity curves show an unusual sigmoidal trend or an exponential decay to a viscous steady state. An original theory of gelation kinetics was developed considering the coarsening of increasingly larger and more stable clusters due to Ostwald ripening and the breakup of clusters that were too large due to the instability of rotating large particles induced by the shear rate. The developed Bounded Ripening Growth model takes into account the trend of the viscosity curves by means of an autocatalytic process with negative feedback on aggregation according to the logistic kinetic equation, in which the constants k1(γ) and k-(ν) are governed by the surface tension and shear rate, respectively. A dimensionless equation based on the difference between the Weber number and the ratio of the inverse kinetic constant to forward constant, accounts for the behavior of the dispersed phase in equilibrium conditions or far from the hydrostatic equilibrium.

Exact solutions based on Segre classification in the f(R) modified theory of gravity

In this article, the Segre classification approach is used to obtain some new solutions for spherically symmetric static spacetime metric corresponding to Segre types [(1, 111)], [1, (111)], [(1, 1)(11)], or [1, 1(11)]. The eigenvalue degeneracy in these situations correlates to timelike and spacelike eigenvectors and their primary null direction, which identify the kind of matter distribution in space and aid in the consideration of novel solutions for the corresponding energy momentum tensor. The isotropic Segre type [(1, 111)] in modified theory provides the Schwarzschild de-Sitter/anti-de-Sitter solutions, whereas in types [1, (111)] and [1, 1(11)] depending upon matter distribution new obtained solutions adhere all the physical conditions and present the viable trends of energy and causality conditions. Moreover, the profiles of the adiabatic index, surface, and gravitational redshift are observed along with the hydrostatic equilibrium using the Tolman–Oppenheimer–Volkoff equation. Additionally, for these types the solution is compared with observational data, and numerical values are calculated for central and surface densities and central pressure of compact star candidates KS1731-260\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$KS 1731-260$$\\end{document} and PSRJ1614-2230\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$PSR J1614-2230$$\\end{document}. Segre type [(1, 1)(11)] relates to a non-null electromagnetic field that corresponds to models with anisotropy in dark energy, which is demonstrated by graphical analysis. Dark energy does not behave as an ordinary matter resulting in the violation of strong energy condition and the causality condition.

Fully higher-order coupling of finite element and level set methods for two-phase flow with a new explicit projection method

This work presents the development of a higher-order finite element method for modeling unsteady, incompressible and inviscid two-phase flows using the level set method. A new explicit projection method is introduced to solve the incompressible Euler equation, where the pressure and the velocity fields are solved separately. To implement high order temporal schemes, the Hodge decomposition in this projection method is not applied to the velocity but to its time derivative. A discontinuous Galerkin discretization is used to solve both the level set and the nonlinear advection terms present in the Euler equations, while a Galerkin discretization is used to solve the pressure Poisson equation. The present method is a fully high order numerical approach, both in space and temporal dimensions. Numerical results provided by several 2D benchmark test cases in naval hydrodynamics (i.e hydrostatic equilibrium, sloshing and dam break) validate the computational code, illustrating the power of the proposed method. Compared to the CFD codes based on low order VOF/finite-volume methods, the present approach demonstrates greater accuracy on coarse meshes, leading to reduced CPU time.

Improved well-balanced AWENO schemes with hydrostatic reconstruction for the Euler equations under gravitational fields

The Euler equations under gravitational fields allow the hydrostatic equilibrium states, which requires that the numerical scheme of the system should also have this characteristic. In our previous work, a well-balanced finite difference conservative AWENO scheme has been constructed to preserve the isothermal equilibrium state accurately (Fu et al., Appl. Numer. Math., 180:1-15, 2022). However, the scheme fails to maintain more complicated hydrostatic equilibrium states, such as the isentropic equilibrium state. In this study, the improved well-balanced AWENO schemes with hydrostatic reconstruction are designed to maintain the general equilibrium states numerically. The key technologies to achieve the well-balanced property are rewriting the source term into an appropriate form based on the equilibrium equation, acting on the conserved variables with an affine-invariant WENO or linearized WENO interpolation operator, employing hydrostatic reconstruction methods to create auxiliary conserved variables, and modifying the numerical flux with auxiliary conserved variables and higher-order conserved terms based on pressure. The proposed well-balanced schemes maintain the equilibrium solution near the machine precision and capture small perturbations of the equilibrium state without spurious oscillations even at the coarse mesh. Furthermore, they can be applicable to the general hydrostatic equilibrium more than the isothermal equilibrium state. Several numerical experiments are tested to show that the developed well-balanced schemes perform well in maintaining equilibrium states and capturing small perturbations.

Color-flavor locked quark stars in Rastall–Rainbow gravity

We investigate the hydrostatic equilibrium configurations of strange stars within the framework of Rastall–Rainbow theory of gravity. These stars are primarily composed of strange quark matter, with its distribution governed by the color-flavor-locked phase of quark matter equation of state (EoS). Using this EoS, we numerically solve the modified Tolman–Oppenheimer–Volkoff equation to obtain different quark star models. We find that the parameters from this modified gravity have specific contributions to the structure of quark stars and introduce new descriptions for massive celestial bodies. Also, the mass–radius relation strongly depends on the parameters from the considered EoS. Furthermore, we analyze the impact of these parameters on the gravitational redshift and compactness of quark stars.

Identifying the influence of design parameters of single-chamber hydrostatic bearing of fuel pump on its main characteristics

The object of this study is hydrostatic processes in the fluid friction bearings of gear-type aviation fuel pumps. The problem of the influence of the design parameters of a single-chamber hydrostatic bearing on its main characteristics was solved. The main characteristics were considered to be the load-bearing capacity and the flow rate of the working fluid. When determining the main characteristics of a single-chamber hydrostatic bearing, the Reynolds and flow balance equations were solved jointly. The resulting diagram of pressure distribution over the working surface of the bearing was used to determine the main characteristics. The influence of the clearance, nozzle diameter, and chamber width on the load-bearing capacity and working fluid flow of a single-chamber hydrostatic bearing was studied. It has been established that as the gap increases, the load-bearing capacity of a single-chamber bearing decreases, and the flow rate of the working fluid increases. As the nozzle diameter increases, the bearing's load-bearing capacity increases. Increasing the width of the chambers leads to an increase in the load-bearing capacity and flow of working fluid through the bearing. When the gap increases from 0.0125 mm to 0.0425 mm, the bearing capacity decreases by 1.156 times. The flow rate of working fluid through the bearing increases by 1.4 times. With an increase in the nozzle diameter from 1.5 mm to 3 mm, the bearing capacity increases slightly by approximately 1.02 times. Increasing the width of the chambers from 4 mm to 8 mm increases the load-bearing capacity by 1.29 times and increases the flow rate of working fluid by 1.4 times. The results show that a single-chamber hydrostatic bearing can provide the required load-bearing capacity by selecting design parameters. The given mathematical dependences could be used for practical calculations of single-chamber hydrostatic bearings

Stress state of the Mars’ and Venus’ interior

It is shown that most of the epicenters of marsquakes are located in the zones of extension and fairly large shear stresses associated with the deviation of Mars from hydrostatic equilibrium. Non-hydrostatic stresses in the interior of Venus are calculated for two types of models: an elastic model and a model with a lithosphere of varying thickness (150–500 km) overlying a weakened layer that has partially lost its elastic properties. Numerical modeling of the system of elastic equilibrium equations for a gravitating planet is carried out with a step of 1°×1° in latitude and longitude up to a depth of 480 km – the first phase transition zone in the mantle. The topography and the gravitational field of the planet are the boundary conditions of the problem. Overall, the level of nonhydrostatic stress on Venus is not very high. On the surface and in the crust, the highest shear stresses are observed in the region of the Maxwell Monte on Ishtar Terra. Beneath the Maxwell Monte, shear stresses in the crust reach 80 MPa and compressive stresses, 125–150 MPa, depending on the model. Tensile stresses around this region are about 20 MPa. The highest tensile stresses occur in the regions beneath structures such as Lavinia Planitia, Sedna Planitia, and Aino Planitia.

Hydrodynamic models of pulsation period evolution in R Hydrae

ABSTRACT Pulsation period evolution during the helium–shell flash in the Mira variable R Hya is investigated using consistent stellar evolution and non-linear stellar pulsation computations. The initial and time-dependent inner boundary conditions for the equations of radiation hydrodynamics describing non-linear stellar oscillations were determined using a grid of TP–AGB model sequences with initial masses on the main sequence 1.5M⊙ ≤ MZAMS ≤ 5.0M⊙ and the initial metallicity Z = 0.014. The setup of initial conditions for hydrodynamic models corresponds to ≈100 yr prior to the maximum of the helium–shell luminosity and ensures that the stellar envelope of the evolution model is under both hydrostatic and thermal equilibrium. Solution of the equations of hydrodynamics allowed us to determine the temporal variation of the pulsation period Π(t) during ≈500 yr. Within this time interval R Hya is a fundamental mode pulsator. The period temporal dependencies Π(t) calculated for the AGB star models at the beginning of the third dredge-up phase and with masses $4.4M_{\odot }\lessapprox M\lessapprox 4.5M_{\odot }$ are in agreement with observational estimates of the period of R Hya obtained during last two centuries. The mean radius of R Hya pulsation models at the end of the XX century ($470 R_{\odot }\lessapprox \bar{R}\lessapprox 490 R_{\odot }$) agrees with observational estimates obtained using the interferometric angular diameter measurements.

A study of an embedding class-I traversable wormhole in Galileon Gravity

In this article, we examine the possibility of the creation of a traversable wormhole geometry in Galileon Gravity (GG). In this gravity, the alternation of gravity is determined by an effective scalar field ϕ, treated as the Galileon field. From a mechanism perspective, GG is developed by covariantizing the decoupling Lagrangian of the DGP (Dvali-Gabadadze-Porrati) model. Now, to achieve our aim, gravitational field equations have been constructed corresponding to a line element, which represents a static and spherically symmetric spacetime. Next, we have modified these field equations in the context of a Morris-Thorne traversable wormhole spacetime. In particular, we have used the ‘Karmarkar condition’ to evaluate the wormhole solutions in a general sense. Further, for a better understanding of our obtained wormhole solutions, 2-dimensional and 3-dimensional diagrams have been displayed. Later, we scrutinized the possible creation of a traversable wormhole based on the essential requirements, that is, the infraction of energy conditions in the neighborhood of the throat region, especially the Null Energy Condition (NEC). For this purpose, we have discussed three cases like Case-I: without the necessity of dark energy (DE), Case-II: with support of phantom-like DE and Case-III: with the SFDM profile of a dark matter (DM) halo. Next, we have examined the hydrostatic equilibrium position corresponding to our obtained wormhole system in GG. Furthermore, to examine the small and sufficient quantity of exotic matter, the possible formation of a traversable wormhole has been explored by a volume integral quantifier. Lastly, we have shown the nature of active mass. Finally, we have concluded that traversable wormhole solutions can also be established in the presence of an effective scalar field without any essential support of exotic matter in the neighborhood of the throat region by GG.

Pathway for a martensitic quartz–coesite transition

An atomistic pathway for a strain-induced subsolidus martensitic transition between quartz and coesite was found by computing the set of the smallest atomic displacements required to transform a quartz structure into a coesite structure. A minimal transformation cell with 24 SiO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\hbox {SiO}_{2}}$$\\end{document} formula units is sufficient to describe the diffusionless martensitic transition from quartz to coesite. We identified two families of invariant shear planes during the martensitic transition, near the {101¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{1}}$$\\end{document}1} and {123¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{3}}$$\\end{document}2} set of planes, in agreement with the orientation of planar defect structures observed in quartz samples which experienced hypervelocity impacts. We calculated the reaction barrier using density functional theory and found that the barrier of 150 meV/atom is pressure invariant from ambient pressure up to 5 GPa, while the mean principal stress limiting the stability of strained quartz is ≈\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\approx$$\\end{document} 2 GPa. The model calculations quantitatively confirm that coesite can be formed in strained quartz at pressures significantly below the hydrostatic equilibrium transition pressure.