Non-hydrostatic

For a discussion of the role of non-hydrostatic pressure in phase equilibria investigations, the effects of applied stress and their consequences in metallurgy, geology, etc. see Cahn (1989). 

Fig. 3. When subjected to non-hydrostatic pressure, the width of the R-lines increases [113, 139], and this has been used to determine the hydrostatic limits of a number of pressure transmitting fluids. An extensive review of ruby at high pressures has recently been written by K. Syassen [142].

Mira variables form an important subgroup of the red giant stars which are typical representatives of stars showing burnt material at their surfaces. Since the photospheres of Miras are not in hydrostatic equilibrium but are characterized by spherically very extended density stratifications, their properties and emitted spectra differ substantially from those of non-Miras, and any attempts to analyse Mira spectra by means of conventional techniques must fail. Non-hydrostatic models are needed for analysis work. 

Chemists and physicists must always formulate correctly the constraints which crystal structure and symmetry impose on their thermodynamic derivations. Gibbs encountered this problem when he constructed the component chemical potentials of non-hydrostatically stressed crystals. He distinguished between mobile and immobile components of a solid. The conceptual difficulties became critical when, following the classical paper of Wagner and Schottky on ordered mixed phases as discussed in chapter 1, chemical potentials of statistically relevant SE s of the crystal lattice were introduced. As with the definition of chemical potentials of ions in electrolytes, it turned out that not all the mathematical operations (9G/9n.) could be performed for SE s of kind i without violating the structural conditions of the crystal lattice. The origin of this difficulty lies in the fact that lattice sites are not the analogue of chemical species (components). 

Let us investigate the steady state behavior of multicomponent crystals exposed to uniform but non-hydrostatic stresses. We first introduce some ideas on the thermodynamics of such solids (which will be discussed in more detail in Chapter 14). Solid state galvanic cells can be used to perform the appropriate experiments. 

Gibbs [J.W. Gibbs (1878)] showed that a non-hydrostatically stressed solid surrounded (Fig. 8-7) by a fluid (in which it is soluble) is entirely determined by the nature and state of the solid through the relation... 

Figure 8-7. Solids under non-hydrostatic stress aud the surrounding Gibbs-fluids L.

The surrounding fluid (Fig. 8-7) serves two purposes 1) it transmits the pressure to stress-load the surface and 2) it allows the surface to equilibrate chemically and thus provides juL in Eqn. (8.61) with physical meaning. Ideally, the Gibbs fluid has a vanishing buffer capacity for the solid so that after a change in an, the fluid becomes resaturated with respect to the solid before a noticeable amount of the solid or its surface dissolves. The key to verify Gibbs relation for solids under non-hydrostatic stress is therefore the existence of such an idealized fluid. 

Figure 8-8. Galvanic cell (schematic) for the determination of the chemical potential difference between surfaces 1 and 2 of non-hydrostatically stressed solids. Cross hatched solid electrolyte ...

We are now able to analyze the demixing of a solid solution (A, B) O ( = MeO) exposed to a non-hydrostatic, uniaxial stress as shown in Figure 8-9. If transport occurs... 

Let us consider a homogeneously, but not hydrostatically, stressed solid which is deformed in the elastic regime and whose structure elements are altogether immobile. If we now isothermally and reversibly add lattice molecules to its different surfaces (with no shear stresses) from the same reservoir, the energy changes are different. This means that the chemical potential of the solid is not single valued, or, in other words, a non-hydrostatically stressed solid with only immobile components does not have a unique measurable chemical potential [J. W. Gibbs (1878)]. 

When all the SE s of a solid with non-hydrostatic (deviatoric) stresses are immobile, no chemical potential of the solid exists, although transport between differently stressed surfaces takes place provided external transport paths are available. Attention should be given to crystals with immobile SE s which contain an (equilibrium) network of mobile dislocations. In these crystals, no bulk diffusion takes place although there may be gradients of the chemical free energy density and, in multicomponent systems, composition gradients (e.g., Cottrell atmospheres [A.H. Cottrell (1953)]). 

The on-line coupled regional meteorology-chemistry model MCCM (Mesoscale climate chemistry model, GreU et al. 2000) has been developed at the IMK-IFU on the basis of the non hydrostatic NCAR/Penn State University mesoscale model... 

A Note on Using the Non-hydrostatic Model AROME as a Driver for the MATCH Model... 

Binary (GRIB) format. From this perspective non-hydrostatic NWP data will appear as inconsistent data. 

Figure 14.1 (middle and bottom panels) show test runs over 18 h for non-hydrostatic driving data (AROME 2.5 km) and hydrostatic data (HIRLAM 5 km). The mass errors with no initialization ranges in both cases from —30 to - - 100% (which means a rather fast deterioration of the mass conservation). 

In Fig. 14.1 (top panel) the impact from the initialization on the horizontal winds is illustrated. The modifications are generally less than 0.1 m/s and over large areas less than 0.02 m/s, both for hydrostatic and non-hydrostatic data. The corrections are thus small and could not be expected to violate the general flow pattern. Note that the corrections seem to be very small in relation to the corresponding errors in Fig. 14.1 (middle and bottom panels), which is a bit misleading as the latter results from integrated errors. 

We have illustrated the need for initialization of non-hydrostatic as well as hydrostatic driving meteorological data for off-line atmospheric chemistry and transport models. The impact on the wind field from initialization is of the same magnitudes for both non-hydrostatic and hydrostatic data (i.e. less than a few dm/s), that indicates that no specific problem concerning initialization for mass conservation of non-hydrostatic data. 

The path towards a coupled meso-scale HARMONIE-chemistry model is not clear yet. There is a possibility to use chemical components that have recently been included in AROME, but they have proved to be quite difficult to use. The Enviro-HIRLAM community is considering two different options for the development of a mesoscale coupled system either the present HARMONIE chemistry code is used and extended with the chemistry and aerosol modules and feedback mechanisms that are already included in Enviro-HIRLAM, or Enviro-HIRLAM can be extended with an updated non-hydrostatic core NH-HIRLAM from Estonia (Room et al. 2007). 

Pinty J-P, Jabouille P (1998) A mixed-phase cloud parameterization for use in mesoscale non-hydrostatic model simulations of a squall line and of orographic precipitations. Proc. conf. of cloud physics, Everett, WA, USA, Amer. Meteor, soc., Aug. 1999, pp 217-220... 

Because structural phase transitions are often ferroelastic or coelastic in character it is essential to have a well-defined stress applied to the crystal at high pressures. In effect, this means that a hydrostatic pressure medium must be used to enclose the crystal. A 4 1 mixture by volume of methanol ethanol remains hydrostatic to just over 10 GPa (Eggert et al. 1992) and is convenient and suitable for many studies. If the sample dissolves in alcohols, then a mixture of pentane and iso-pentane which remains hydrostatic to 6 GPa (Nomura et al. 1982), or a solidified gas such as N2, He, or Ar can be employed. Water appears to remain hydrostatic to about 2.5 GPa at room temperature, just above the phase transition from ice-VI to ice-VII (Angel, unpublished data). The solid pressure media such as NaCl or KCl favoured by spectroscopists are very non-hydrostatic even at pressures below 1 GPa and have been shown to displace phase transitions by at least several kbar (e g. Sowerby and Ross 1996). Similarly, the fluorinert material used in many neutron diffraction experiments because of its low neutron scattering power becomes significantly non-hydrostatic at -1.3 GPa. Decker et al. (1979) showed that the ferroelastic phase transition that occurs at 1.8 GPa in lead phosphate under hydrostatic conditions is not observed up to 3.6 GPa when fluorinert was used as the pressure medium. At pressures in excess of the hydrostatic limit of the solidified gas and fluid... 

Chai M, Brown JM (1996) Effects of non-hydrostatic stress on the R lines of raby single crystals. Geophys Res Letts 23 3539-3542... 

Coen and Clark [128] has coupled a fire model into a three-dimensional non-hydrostatic terrain-following numerical mesoscale model developed at the US National Center for Atmospheric Research, Boulder, CO. The model includes rain and cloud physics. Calculations predict the growth and spread of a fire line moving across a two dimensional small Gaussian hill (height 200 m, half-width 300 m) for a wind speed of 3 m/s, and a stable atmospheric lapse rate (10°C/km). The head of the fire propagated quickly uphill in the direction of the environmental wind. Once the fire reaches the top of the hill, the updrafts tend to inhibit the forward movement of the fire front, and the fire spreads faster laterally in the lee of the hill. 

In Norway, the non-hydrostatic MM5 model (Grell et al., 1994 [234]) is nested with the HIRLAM NWP model (Berge et al., 2002 [58]). The latter model is operated on a 10 km horizontal resolution for North-Western Europe. A domain with a resolution of 3 km has been set up for the region around the city of Oslo in which MM5 is one-way nested with HIRLAM. A two-way nesting takes place between the 3 km resolution... 

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