Showing papers on "Hydrostatic equilibrium published in 2003"

Dynamically balanced absolute sea level of the global ocean derived from near‐surface velocity observations

Abstract: [1] The 1992–2002 time-mean absolute sea level distribution of the global ocean is computed for the first time from observations of near-surface velocity. For this computation, we use the near-surface horizontal momentum balance. The velocity observed by drifters is used to compute the Coriolis force and the force due to acceleration of water parcels. The anomaly of horizontal pressure gradient is derived from satellite altimetry and corrects the temporal bias in drifter data distribution. NCEP reanalysis winds are used to compute the force due to Ekman currents. The mean sea level gradient force, which closes the momentum balance, is integrated for mean sea level. We find that our computation agrees, within uncertainties, with the sea level computed from the geostrophic, hydrostatic momentum balance using historical mean density, except in the Antarctic Circumpolar Current. A consistent horizontally and vertically dynamically balanced, near-surface, global pressure field has now been derived from observations.

Dynamic Cores in Hydrostatic Disguise

Abstract: We discuss the column density profiles of "cores" in three-dimensional smoothed particle hydrodynamics (SPH) numerical simulations of turbulent molecular clouds. The SPH scheme allows us to perform a high spatial resolution analysis of the density maxima (cores) at scales between ~0.003 and 0.3 pc. We analyze simulations in three different physical conditions: large-scale driving (LSD), small-scale driving (SSD), and random Gaussian initial conditions without driving (GC), each one at two different time steps: just before self-gravity is turned on (t0) and when gravity has been operating such that 5% of the total mass in the box has been accreted into cores (t1). For this data set, we perform Bonnor-Ebert fits to the column density profiles of cores found by a clump-finding algorithm. We find that, for the particular fitting procedure we use, 65% of the cores can be matched to Bonnor-Ebert (BE) profiles, and of these, 47% correspond to stable equilibrium configurations with ξmax < 6.5, even though the cores analyzed in the simulations are not in equilibrium but instead are dynamically evolving. The temperatures obtained with the fitting procedure vary between 5 and 60 K (in spite of the simulations being isothermal, with T = 11.3 K), with the peak of the distribution being at T = 11 K and most clumps having fitted temperatures between 5 and 30 K. Central densities obtained with the BE fit tend to be smaller than the actual central densities of the cores. We also find that for the LSD and GC cases, there are more BE-like cores at t0 than at t1 with ξmax ≤ 20, while in the case of SSD, there are more such cores at t1 than at t0. We interpret this as a consequence of the stronger turbulence present in the cores of run SSD, which prevents good BE fits in the absence of gravity, and delays collapse in its presence. Finally, in some cases we find substantial superposition effects when we analyze the projection of the density structures, even though the scales over which we project are small (0.18 pc). As a consequence, different projections of the same core may give very different values of the BE fits. Finally, we briefly discuss recent results claiming that Bok globule B68 is in hydrostatic equilibrium, stressing that they imply that this core is unstable by a wide margin. We conclude that fitting BE profiles to observed cores is not an unambiguous test of hydrostatic equilibrium and that fit-estimated parameters such as mass, central density, density contrast, temperature, or radial profile of the BE sphere may differ significantly from the actual values in the cores.

Thermal Instability in Clusters of Galaxies with Conduction

Abstract: We consider a model of galaxy clusters in which the hot gas is in hydrostatic equilibrium and maintains energy balance between radiative cooling and heating by thermal conduction. We analyze the thermal stability of the gas using a Lagrangian perturbation analysis. For thermal conductivity at the level of ~20%-40% of Spitzer conductivity, consistent with previous estimates for cluster gas, we find that the growth rate of the most unstable global radial mode is ~6-9 times lower than the growth rate of local isobaric modes at the cluster center in the absence of conduction. The growth time in typical clusters is ~2-5 Gyr, which is comparable to the time since the last major merger episode, when the gas was presumably well mixed. Thus, we suggest that thermal instability is not dynamically significant in clusters, provided that there is an adequate level of thermal conduction. On the other hand, if the heating of the gas is not the result of thermal conduction or any other diffusive process such as turbulent mixing, then the thermal instability has a growth time under a gigayear in the central regions of the cluster and is a serious threat to equilibrium. We also analyze local nonradial modes and show that the Lagrangian technique leads to the same dispersion relation as the Eulerian approach, provided that clusters are initially in strict thermal equilibrium. Because cluster gas is convectively stable, nonradial modes always have a smaller growth rate than equivalent radial modes.

Modeling Intracluster Gas in Triaxial Dark Halos: An Analytic Approach

Abstract: We present the first physical model for the nonspherical intracluster gas distribution in hydrostatic equilibrium under the gravity of triaxial dark matter halos. Adopting the concentric triaxial density profiles of dark halos with constant axis ratios proposed by Jing and Suto in 2002, we derive an analytical expression for the triaxial halo potential on the basis of perturbation theory and find the hydrostatic solutions for the gas density and temperature profiles in both isothermal and polytropic equations of state. The resulting isopotential surfaces are well approximated by triaxial ellipsoids with the eccentricities dependent on the radial distance. We also find a formula for the eccentricity ratio between the intracluster gas and the underlying dark halo. Our results allow one to determine the shapes of the underlying dark halos from the observed intracluster gas through X-ray and/or Sunyaev-Zeldovich effect clusters.

Turbulent Mixing in Clusters of Galaxies

Abstract: We present a spherically symmetric, steady state model of galaxy clusters in which radiative cooling from the hot gas is balanced by heat transport through turbulent mixing. We assume that the gas is in hydrostatic equilibrium and describe the turbulent heat diffusion by means of a mixing length prescription with a dimensionless parameter αmix. Models with αmix ~ 0.01-0.03 yield reasonably good fits to the observed density and temperature profiles of cooling core clusters. Making the strong simplification that αmix is time independent and that it is roughly the same in all clusters, the model reproduces remarkably well the observed scalings of X-ray luminosity, gas mass fraction, and entropy with temperature. The break in the scaling relations at kT ~ 1-2 keV is explained by the break in the cooling function at around this temperature, and the entropy floor observed in galaxy groups is reproduced naturally.

A fully hydrodynamic model for three‐dimensional, free‐surface flows

Abstract: The hydrostatic pressure assumption has been widely used in studying water movements in rivers, lakes, estuaries, and oceans. While this assumption is valid in many cases and has been successfully used in numerous studies, there are many cases where this assumption is questionable. This paper presents a three-dimensional, hydrodynamic model for free-surface flows without using the hydrostatic pressure assumption. The model includes two predictor-corrector steps. In the first predictor-corrector step, the model uses hydrostatic pressure at the previous time step as an initial estimate of the total pressure field at the new time step. Based on the estimated pressure field, an intermediate velocity field is calculated, which is then corrected by adding the non-hydrostatic component of the pressure to the estimated pressure field. A Poisson equation for non-hydrostatic pressure is solved before the second intermediate velocity field is calculated. The final velocity field is found after the free surface at the new time step is computed by solving a free-surface correction equation

A non-hydrostatic numerical model for calculating free-surface stratified flows

Abstract: A three-dimensional non-hydrostatic numerical model for simulation of the free-surface stratified flows is presented. The model is a non-hydrostatic extension of free-surface primitive equation model with a general vertical coordinate and horizontal orthogonal curvilinear coordinates. The model equations are integrated with mode-splitting technique and decomposition of pressure and velocity fields on hydrostatic and non-hydrostatic components. The model was tested against laboratory experiments on the steep wave transformation over the longshore bar, solitary wave impact on the vertical wall, the collapse of the mixed region in the thin pycnocline, mixing in the lock-exchange flows and water exchange through the sea strait. The agreement is generally fair.

Water table waves in an unconfined aquifer: Experiments and modeling

Abstract: [1] Comprehensive measurements are presented of the piezometric head in an unconfined aquifer during steady, simple harmonic oscillations driven by a hydrostatic clear water reservoir through a vertical interface. The results are analyzed and used to test existing hydrostatic and nonhydrostatic, small-amplitude theories along with capillary fringe effects. As expected, the amplitude of the water table wave decays exponentially. However, the decay rates and phase lags indicate the influence of both vertical flow and capillary effects. The capillary effects are reconciled with observations of water table oscillations in a sand column with the same sand. The effects of vertical flows and the corresponding nonhydrostatic pressure are reasonably well described by small-amplitude theory for water table waves in finite depth aquifers. That includes the oscillation amplitudes being greater at the bottom than at the top and the phase lead of the bottom compared with the top. The main problems with respect to interpreting the measurements through existing theory relate to the complicated boundary condition at the interface between the driving head reservoir and the aquifer. That is, the small-amplitude, finite depth expansion solution, which matches a hydrostatic boundary condition between the bottom and the mean driving head level, is unrealistic with respect to the pressure variation above this level. Hence it cannot describe the finer details of the multiple mode behavior close to the driving head boundary. The mean water table height initially increases with distance from the forcing boundary but then decreases again, and its asymptotic value is considerably smaller than that previously predicted for finite depth aquifers without capillary effects. Just as the mean water table over-height is smaller than predicted by capillarity-free shallow aquifer models, so is the amplitude of the second harmonic. In fact, there is no indication of extra second harmonics ( in addition to that contained in the driving head) being generated at the interface or in the interior.

Gravitational Collapse of Filamentary Magnetized Molecular Clouds

Abstract: We develop models for the self-similar collapse of magnetized isothermal cylinders. We find solutions for the case of a fluid with a constant toroidal flux-to-mass ratio (Γ = constant) and the case of a fluid with a constant gas to magnetic pressure ratio (β = constant). In both cases, we find that a low magnetization results in density profiles that behave as ρ ∝ r-4 at large radii, and at high magnetization we find density profiles that behave as ρ ∝ r-2. This density behavior is the same as for hydrostatic filamentary structures, suggesting that density measurements alone cannot distinguish between hydrostatic and collapsing filaments—velocity measurements are required. Our solutions show that the self-similar radial velocity behaves as vr ∝ r during the collapse phase, and that unlike collapsing self-similar spheres, there is no subsequent accretion (i.e., expansion-wave) phase. We also examine the fragmentation properties of these cylinders and find that in both cases, the presence of a toroidal field acts to strengthen the cylinder against fragmentation. Finally, the collapse timescales in our models are shorter than the fragmentation timescales. Thus, we anticipate that highly collapsed filaments can form before they are broken into pieces by gravitational fragmentation.

Remarks on the derivation of the hydrostatic Euler equations

Abstract: The motion of an inviscid incompressible fluid between two horizontal plates is studied in the limit when the plates are infinitesimally close. The convergence of the solutions of the Euler equations to those of their formal ‘hydrostatic’ limit can be established in the case when the initial velocity field satisfies a local Rayleigh conditions. This result, originally obtained by Grenier through weighted energy estimates based on Arnold's stability analysis of the Euler equations, is proven here by a more straightforward method even closer to Arnold's method.

Shallow-water theory for arbitrary slopes of the bottom

Abstract: A new shallow-water theory valid for arbitrary bottom slope, due to Bouchut et al. (2003), is derived systematically by a scaling method. The fact that the pressure is hydrostatic, and the form of the velocity parallel to the bottom, are consequences of the scaling method, and need not be assumed.

Atmospheric gravity perturbations measured by a ground-based interferometer with suspended mirrors

Abstract: In this paper, we discuss the possibility of making geophysical measurements using the large-scale laser interferometrical gravitational wave antenna. An interferometer with suspended mirrors can be used as a gradiometer measuring variations of an angle between gravity force vectors acting on the spatially separated suspensions. We analyse the restrictions imposed by the atmospheric noises on the feasibility of such measurements. Two models of the atmosphere are invoked: a quiet atmosphere with a hydrostatic coupling of pressure and density and a dynamic model of moving region of the density anomaly (cyclone). Both models lead to similar conclusions up to numerical factors. Besides the hydrostatic approximation, we use a model of turbulent atmosphere with the pressure fluctuation spectrum ~f−7/3 to explore the Newtonian noise in a higher frequency domain (up to 10 Hz) predicting the gravitational noise background for modern gravitational wave detectors. Our estimates show that this could pose a serious problem for realization of such projects. Finally, angular fluctuations of spatially separated pendula are investigated via computer simulation for some realistic atmospheric data giving the level estimate ~10−11 rad Hz−1/2 at frequency ~10−4 Hz. This looks promising for the possibility of the measurement of weak gravity effects such as Earth inner core oscillations.

Gravitational Collapse of Filamentary Magnetized Molecular Clouds

Abstract: We develop models for the self-similar collapse of magnetized isothermal cylinders. We find solutions for the case of a fluid with a constant toroidal flux-to-mass ratio (Gamma_phi=constant) and the case of a fluid with a constant gas to magnetic pressure ratio (beta=constant). In both cases, we find that a low magnetization results in density profiles that behave as rho ~ r^{-4} at large radii, and at high magnetization we find density profiles that behave as rho ~ r^{-2}. This density behaviour is the same as for hydrostatic filamentary structures, suggesting that density measurements alone cannot distinguish between hydrostatic and collapsing filaments--velocity measurements are required. Our solutions show that the self-similar radial velocity behaves as v_r ~ r during the collapse phase, and that unlike collapsing self-similar spheres, there is no subsequent accretion (i.e. expansion-wave) phase. We also examine the fragmentation properties of these cylinders, and find that in both cases, the presence of a toroidal field acts to strengthen the cylinder against fragmentation. Finally, the collapse time scales in our models are shorter than the fragmentation time scales. Thus, we anticipate that highly collapsed filaments can form before they are broken into pieces by gravitational fragmentation.

The deep-atmosphere Euler equations with a mass-based vertical coordinate

Abstract: The terrain-following hydrostatic pressure-coordinate system for the shallow-atmosphere Euler equations is generalized for deep atmospheres using a mass-based vertical coordinate. A consequent benefit is that an existing (shallow atmosphere) hydrostatic primitive-equations model, which uses a pressure-based terrain-following vertical coordinate, could be modified for non-hydrostatic deep-atmosphere applications, without the need to change the scientific and computing infrastructure substantially in which it is embedded. © Crown copyright, 2003. Royal Meteorological Society

On the Pressure Field in the Slope Wind Layer

Abstract: It has been suggested by some authors that the momentum equation for thermally driven slope flow should contain a horizontal pressure gradient term, in addition to the buoyancy term. It is shown that this suggestion is incorrect and leads to a spurious increase in along-slope forcing unless the vertical component of the perturbation pressure gradient is included as well. Along-slope accelerations due to the horizontal and vertical perturbation pressure gradients cancel each other exactly if the temperature perturbation is constant along the slope. Based on the concept of hydrostatic equilibrium perpendicular to the slope, the error associated with neglecting the vertical component of the pressure gradient, and the error due to the assumption of vertical hydrostatic equilibrium are evaluated. A revised conceptual diagram of the relationship between buoyancy and pressure forces within the slope wind layer is presented.

Gravity wave breaking in two-layer hydrostatic flow

Abstract: To better understand mountain-induced gravity wave breaking and potential vorticity generation in the troposphere, a two-layer hydrostatic flow over a three-dimensional Witch-of-Agnesi type of mountain is investigated. It is suggested that a two-layer model is the simplest model in which the partitioning of upper- and lower-level wave breaking and dissipation can be studied. High-resolution shallow water model runs are carried out with unsheared upstream flow and a wide variety of mountain heights. A regime diagram is constructed, in which gravity wave breaking is classified based on shock number, location, and type. It is demonstrated that different types of shocks identified in the numerical simulations can be consistently described using a shock regime diagram, derived from viscous shock theory. Four curious shock properties are shown to influence orographic flow: the steepening requirement, the tendency for external jumps to amplify shear, the bifurcation in external jumps, and the “double sh...

Hydraulic torque vectoring differential

Abstract: A hydraulic torque vectoring differential includes two epicyclic gear sets (62, 65) and two variable displacement hydrostatic units (90, 92). Each hydrostatic unit is coupled to a reaction member (80, 82) of one of each of the epicyclic gear sets, each of which also has a first gear element (58) coupled to an input drive shaft (53) for power input from a prime mover of said vehicle and a third gear element (59) coupled to an output shaft (60) operatively driving the wheels of the vehicle. The hydrostatic units are hydraulically coupled so that the hydraulic fluid pressurized in one hydrostatic unit drives the other hydrostatic unit, and fluid pressurized in the other hydrostatic unit drives the one hydrostatic unit. A control system (105) controls the displacement of the variable displacement hydrostatic units. Power from the prime mover flows primarily through the epicyclic gear sets to the output shafts, and only differential power is passed through the hydrostatic units, thereby isolating the hydraulic units from the primary power flow and making use of low displacement hydrostatic units possible for said differential power flow through said differential. The desired torque distribution between the two wheels is determined by existing conventional computer controls based on inputs from known traction sensors.

On the active gravitational mass of a non-spherical source leaving hydrostatic equilibrium

Abstract: We obtain an expression for the active gravitational mass (Tolman) of a source of the γ metric, just after its departure from hydrostatic equilibrium, on a time scale of the order of (or smaller than) the hydrostatic time scale. It is shown that for very compact sources, even arbitrarily small departures from sphericity produce a significant decrease (increase) in the value of an active gravitational mass of collapsing (expanding) spheres, with respect to its value in equilibrium, enhancing thereby the stability of the system.

The equilibrium form of a rotating earth with an elastic shell

Abstract: SUMMARY The equilibrium form of the Earth is generally computed using a hydrostatic theory that assumes a rotating, inviscid planet. We compute the perturbation to this equilibrium form due to the presence of a thin elastic lithospheric shell using viscoelastic Love number theory. The thin shell acts to reduce the flattening of the equilibrium form relative to the value obtained from the traditional hydrostatic calculation. Our results indicate that current estimates of the excess non-hydrostatic flattening of the Earth, defined as the discrepancy between the observed and hydrostatic forms, may therefore be underestimating the actual departure of the observed form from its equilibrium state. This conclusion may be important for viscous flow models of mantle convection, which are commonly constrained to fit the non-hydrostatic flattening. For completeness, we also adopt the Love number formulation to estimate the excess flattening associated with the gradual slowing of the Earth's rotation. Our predictions of the fossil rotational bulge confirm the widespread view that this effect is small for reasonable mantle viscosity profiles.

A Numerical Study of Relativistic Fluid Collapse

Abstract: We investigate the dynamics of self-gravitating, spherically-symmetric distributions of fluid through numerical means. In particular, systems involving neutron star models driven far from equilibrium in the strong-field regime of general relativity are studied. Hydrostatic solutions of Einstein's equations using a stiff, polytropic equation of state are used for the stellar models. Many of the scenarios we examine involve highly-relativistic flows that require improvements upon previously published numerical methods to simulate. Here our particular focus is on the physical behavior of the coupled fluid-gravitational system at the threshold of black hole formation--so-called black hole critical phenomena. To investigate such phenomena starting from conditions representing stable stars, we must drive the star far from its initial stable configuration. We use one of two different mechanisms to do this: setting the initial velocity profile of the star to be in-going, or collapsing a shell of massless scalar field onto the star. Both of these approaches give rise to a large range of dynamical scenarios that the star may follow. These scenarios have been extensively surveyed by using different initial star solutions, and by varying either the magnitude of the velocity profile or the amplitude of the scalar field pulse. In addition to illuminating the critical phenomena associated with the fluid collapse, the resulting phase diagram of possible outcomes provides an approximate picture of the stability of neutron stars to large, external perturbations that may occur in nature.

A two-parameter framework to describe effects of constraint loss on cleavage fracture and implications for failure assessments of cracked components

Abstract: This study builds upon the J-Q approach to characterize constraint effects on cleavage fracture behavior of cracked structural components. Discussions emphasize features of current two-parameter fracture methodologies which extend the limits of applicability of single parameter fracture approaches when LSY effects prevail. Inclusion of the second parameter (Q) in failure assessment procedures leads to the construction of experimentally derived fracture toughness loci, rather than conventional, single-valued definitions of toughness. The plan of the article is as follows. First, the notion of crack tip constraint and its connection with SSY reference fields is introduced. This is followed by a brief description of the J-Q theory to define the hydrostatic parameter Q. The paper then addresses representative numerical solutions which provide J-Q trajectories for common fracture specimens under bend and tensile loading, including deep and shallow crack SE(B) and SE(T) specimens. These analyses, when taken together with previous works, provide a fairly extensive body of results against which the robustness of the J-Q methodology can be weighed.

Storage and Equilibrium of Toroidal Magnetic Fields in the Solar Tachocline: A Comparison between MHD Shallow-Water and Full MHD Approaches

Abstract: Recently Dikpati & Gilman have shown, using a shallow-water model of the solar tachocline that allows the top surface to deform, that a tachocline with the observed broad differential rotation and a strong toroidal field is prolate. A strong toroidal field ring requires extra mass on its poleward side to provide a hydrostatic latitudinal pressure gradient to balance the poleward curvature stress. In a parallel study using a different approach, Rempel, Schussler, & Toth have shown that such a latitudinal pressure gradient is found in a strongly subadiabatic stratification, whereas a weakly subadiabatic stratification leads to a complementary equilibrium state of the overshoot tachocline in which the magnetic curvature stress is balanced by a prograde rotational jet inside the toroidal ring. We show that the shallow-water model with height deformation is a first-order approach to the equilibrium state found by Rempel, Schussler, & Toth for a strongly subadiabatic stratification. We also show that the shallow-water model can be generalized to allow for the equilibrium state found for a weakly subadiabatic stratification by suppressing the shell deformation associated with the toroidal field and allowing the differential rotation to be modified.

Calculation of Pressure and Density in Solar Power Plant Chimneys

Abstract: This technical brief develops calculation methods for the pressure drop in very tall chimneys, as in solar chimney power plants. The methods allow for density and flow area change with height, for wall friction and internal bracing drag. It presents equations for the vertical pressure and density distributions in terms of Mach number One of these is a generalization of the adiabatic pressure lapse ratio equation to include flow at small Mach numbers. The other is analogous to the hydrostatic relationship between pressure, density, and height, but extends it to small Mach numbers. Its integration leads to an accurate value of the average density in the chimney.

Analysis of a jet stream induced gravity wave associated with an observed ice cloud over Greenland

Abstract: A polar stratospheric ice cloud (PSC type II) was observed by airborne lidar above Greenland on 14 January 2000. Is was the unique observation of an ice cloud over Greenland during the SOLVE/THESEO 2000 campaign. Mesoscale simulations with the hydrostatic HRM model are presented which, in contrast to global analyses, are capable to produce a vertically propagating gravity wave that induces the low temperatures at the level of the PSC afforded for the ice formation. The simulated minimum temperature is ~8 K below the driving analyses and ~3 K below the frost point, exactly coinciding with the location of the observed ice cloud. Despite the high elevations of the Greenland orography the simulated gravity wave is not a mountain wave. Analyses of the horizontal wind divergence, of the background wind profiles, of backward gravity wave ray-tracing trajectories, of HRM experiments with reduced Greenland topography and of several instability diagnostics near the tropopause level provide consistent evidence that the wave is emitted by the geostrophic adjustment of a jet instability associated with an intense, rapidly evolving, anticyclonically curved jet stream. In order to evaluate the potential frequency of such non-orographic polar stratospheric cloud events, an approximate jet instability diagnostic is performed for the winter 1999/2000. It indicates that ice-PSCs are only occasionally generated by gravity waves emanating from an unstable jet.

Seismic velocity, anisotropy, and fluid pressure in the Barbados accretionary wedge from an offset vertical seismic profile with seabed sources

Abstract: [1] The state of compaction and fluid pressure in the Barbados accretionary wedge near its toe, at Ocean Drilling Program Site 949, were investigated by modeling travel times of seismic waves from ocean bottom shots to a borehole geophone array. The model, constrained by a three-dimensional seismic survey and well logs, shows (1) a velocity gradient of about 1–1.25 s−1 in the uppermost 180–230 m of the wedge; (2) a zone of variable, but no net change in, velocity between 230 and 350 m depth; (3) a low-velocity zone 40–50 m thick just above the decollement at 391 m; and (4) a displacement of the low-velocity zone by thrust faults. Pore fluid pressure sections derived from P wave velocity show that the upper half of the wedge is normally pressured while the lower half is overpressured. The ∼160 m thick, underconsolidated basal zone shows anisotropy, which increases downward. The lowest 40–50 m has velocity varying (1) azimuthally (3%), being fastest in the direction of plate convergence, and (2) in the vertical plane (2–5%), horizontal faster than vertical. After correction for the effect of anisotropy in the derivation of effective stress from seismic velocity the calculated pore fluid pressure ratio λ does not exceed 0.9 and in the lowest 40–50 m of the basal zone, is between 0.71 and 0.82, with λ* [(fluid pressure − hydrostatic)/(lithostatic pressure − hydrostatic)] between 0.5 and 0.65, in accordance with in situ measurements of fluid pressure in the decollement zone beneath. These indicate that the accretionary wedge is stronger and less overpressured than was previously supposed.

Stability of leap-frog constant-coefficients semi-implicit schemes for the fully elastic system of Euler equations. Flat-terrain case

Abstract: The aim of this paper is to investigate the response of this system/scheme in terms of stability in presence of explicitly treated residual terms, as it inevitably occurs in the reality of NWP. This sudy is restricted to the impact of thermal and baric residual terms (metric residual terms linked to the orography are not considered here). It is shown that conversely to what occurs with Hydrostatic Primitive Equations, the choice of the prognostic variables used to solve the system in time is of primary importance for the robustness with Euler Equations. For an optimal choice of prognostic variables, unconditionnally stable schemes can be obtained (with respect to the length of the time-step), but only for a smaller range of reference states than in the case of Hydrostatic Primitive Equations. This study also indicates that: (i) vertical coordinates based on geometrical height and on mass behave similarly in terms of stability for the problems examined here, and (ii) hybrid coordinates induce an intrinsic instability, the practical importance of which is however not completely elucidated in the theoretical context of this paper.

Three-dimensional hydrodynamic model for free surface flow

Abstract: A three-dimensional mathematical model for free surface flow based on the Reynolds-averaged Navier-Stokes equations is developed in a sigma co-ordinate system. The time-splitting method is used to separate advection and diffusion terms from the pressure terms in the governing equations. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method, while the hydrostatic pressure equations are solved very efficiently by an Alternating Direction Implicit (ADI) scheme. The convection terms are discretized by the Roe's scheme of second-order accuracy. A staggered mesh is used. The model is tested against available analytical solutions and experimental data. Good agreement has been achieved.