The shape of pebbles

A theory of pebble erosion is presented, based on the assumption that the rate of erosion at a point on the surface is a function Vof the curvature there. It is proved that for physically reasonable functions V,the sphere is the only shape of pebble which can maintain its proportions as it wears away. An argument is given which leads to a particular form for the function Vand a few qualitative consequences of this form are indicated. The surface of the pebble at time tmay be described using spherical polar coordinates θ, Φ by the radius function r (θ, Φ, t). This function is given by a highly nonlinear partial differential equation. However, in the case of the erosion of a deformed sphere, when terms which are of second order or higher in the deformation are neglected, the equation becomes linear and is a version of the diffusion equation. The stability of the spherical shape against deformations of the various harmonic types is then easily analyzed.     

Some properties of the development of the perturbed zone and shock preceding a coronal mass ejection

SOHO/LASCO C2 and C3 data have been used to carry out a detailed study of the perturbed zone and shock that form as a coronal mass ejection (CME) moves away from the Sun, as a result of its interaction with the ambient solar wind. The event of January 4, 2002 is used as an example. The perturbed zone is most extensive along the direction of propagation of the CME, decreases away from this direction, and reaches its minimum values perpendicular to this direction. The mass of the perturbed zone is ≥0.1 of the total mass of the CME. The condition for the formation of a shock preceding the CME (in the direction of propagation of the CME) is V − V _SW > V _A , where V, V _SW , and V _A are the CME, solar wind, and Alfvén velocities, respectively. Perpendicular to the CME axis, at distances of ≈4–6R _⊙ fromthe center of the Sun, the condition for the formation of shock is V/2 > V _A .     

Post-injection spreading and trapping of CO2 in saline aquifers: impact of the plume shape at the end of injection

We use an analytical model for the post-injection spreading of a plume of CO₂ in a saline aquifer under the action of buoyancy and capillary trapping to show that the spreading behavior is at all times strongly influenced by the shape of the plume at the end of the injection period. We solve the spreading equation numerically and confirm that, at late times, the volume of mobile CO₂ is given by existing asymptotic analytical solutions. The key parameters governing plume spreading are the mobility ratio, M, and the capillary trapping number, Γ—the former sets the shape of the plume at the end of the injection period, and the latter sets the amount of trapping. As a quantitative measure of the dependence of the spreading behavior on the initial shape, we use a volume ratio. That is, we evolve the plume from a true end-of-injection initial shape and also from an idealized “step” initial shape, and we take the ratio of these mobile plume volumes in the asymptotic regime. We find that this volume ratio is a power-law in M, where the exponent is governed exclusively by Γ. For conditions that are representative of geologic CO₂ sequestration, the ratio of mobile volumes between “true” and “step” initial plume shapes can be 50% or higher.     

Deformability characteristics of a rock mass under true-triaxial stress compression

In this paper an experimental study was planned on rock mass model with three joint sets under triaxial and true-triaxial stress states to assess the influence of joint geometry and stress ratios on deformational behaviour of rock mass. The physical models were composed of three continuous orthogonal joint sets in which joint set-I was inclined at angle θ=0°, 20°, 40°, 60°, 80° and 90° with x-axis, joint set-II was produced at staggering s=0.5 and joint set-III was kept always vertical. Thus, rock mass models with medium interlocked smooth joints (ϕ _j =36.8°) were simulated under true triaxial compression (σ₁>σ₂>σ₃). Modulus of rock mass shows anisotropy with joint inclination θ which diminishes with increase in σ₂/σ₃ ratio. The rock mass at θ=60° shows the highest modulus enhancement (599.9%) whereas it is minimum (32.3%) at θ=90°. Further two empirical expressions for estimation of deformation modulus were suggested based on experimental results, which were developed by incorporating two basic concepts, e.g. Janbu’s coefficients and joint factor, J _f.     

On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere

Covariance and variogram functions have been extensively studied in Euclidean space. In this article, we investigate the validity of commonly used covariance and variogram functions on the sphere. In particular, we show that the spherical and exponential models, as well as power variograms with 0<α≤1, are valid on the sphere. However, two Radon transforms of the exponential model, Cauchy model, the hole-effect model and power variograms with 1<α≤2 are not valid on the sphere. A table that summarizes the validity of commonly used covariance and variogram functions on the sphere is provided.     

P-V-T equation of state of lawsonite

A pressure-volume-temperature data set has been obtained for lawsonite [CaAl₂Si₂O₇(OH)₂.H₂O], using synchrotron X-ray diffraction and an externally heated diamond anvil cell. Unit-cell volumes were measured to 9.4 GPa and 767 K by angle dispersive X-ray diffraction using imaging plates. Phase changes were not observed within this pressure-temperature range, and lawsonite compressed almost isotropically at constant temperature. The P-V-T data have been analyzed using a Birch- Murnaghan equation of state and a linear equation of state expressed as β=–1/V₀ (∂V/∂P) _T . At room temperature, the derived equation of state parameters are: K ₀=124.1 (18) GPa K'₀ set to 4) and β^–1=142.0(24) GPa, respectively. Our results are intermediate between previously reported measurements. The high-temperature data show that the incompressibility of lawsonite decreases with increasing temperature to ∼500 K and then increases above. Hence, the second order temperature derivative of the bulk modulus is taken into account in the equation of state; a fit of the volume data yields K ₀=123.9(18) GPa, (∂K/∂T)_P=–0.111(3) GPa K^–1, (∂² K/∂T ²)_P=0.28(6) 10^–3 GPa K^–2, α₀=3.1(2) 10^–5 K^–1, assuming K'₀=4. Received: 2 June 1998 / Revised, accepted: 12 Ocotber 1998     

The Direct Shear Strength and Dilatancy of Sand–gravel Mixtures

A total of 87 direct shear tests in a large direct shear-box apparatus have been used to investigate the strength and dilatancy of sand–gravel mixtures. This paper focuses on the differences in behaviour between a silica sand (yellow Leighton Buzzard sand) and sand–gravel mixtures obtained by adding fractions of two kinds of gravel to the sand. The purpose is to find a relation between the grain-size characteristics of the materials and the shearing resistance. Experimental results are analysed in terms of the frictional and dilatant contributions to the strength of mixtures as a function of their relative density, and are compared with dilatancy theories and empirical equations. The addition of gravel to the mixtures, even at low fractions (less than 0.1 by volume), causes an increase in peak friction angle (Φ ′_peak) which results both from higher dilatancy at failure (ψ_max) and higher constant volume friction angle (Φ ′_cv). Use of the minimum voids ratio (e_min) of the materials allows the data for the two families of mixtures to be normalized and interpreted in terms of Φ ′_cv and the ratio (Φ ′_peak − Φ ′_cv)/ψ_max. The relationships between relative density (D_r), ψ_max and Φ ′_peak − Φ ′_cv are only partly explained on a physical basis, so we develop empirical equations to predict the peak shear resistance of sand–gravel mixtures (up to gravel contents of 0.5) on the basis of easily measurable quantities. Such equations constitute a practical tool to overcome the problems arising from the impracticality of testing coarse material in the standard shear-box apparatus.     

The elastic constants of monoclinic single-crystal chrome-diopside to 1,300 K

Values of the complete adiabatic elastic tensor for single-crystal chrome-diopside (a monoclinic pyroxene mineral) are presented from 298 to 1,300 K. The data were obtained using resonant ultrasound spectroscopy (RUS). They are the first published results for the temperature T dependences of the 13 individual elastic constants C _ij of any clinopyroxene mineral. Each C _ij is appropriately described by a linear function in T throughout the range of T. Values for each (∂C _ij /∂T) _P in GPa K⁻¹ are as follows: C ₁₁, −0.0291; C ₂₂, −0.0248; C ₃₃, −0.0179; C ₄₄, −0.0103; C ₅₅, −0.0077; C ₆₆, −0.0152; C ₁₂, −0.0119; C ₁₃, −0.0064; C ₂₃, 0.0000; C ₁₅, 0.0025; C ₂₅, 0.0022; C ₃₅, −0.0046; and C ₄₆, 0.0026. Values of (∂M/∂T) _P in GPa K⁻¹, where M represents an isotropic bulk property calculated from the C _ij data, are as follows: adiabatic bulk modulus K _S , −0.0123; isothermal bulk modulus K _T , −0.0178; and shear modulus G, −0.00998. Some diopside derivatives, notably (∂K _S /∂T) _P , (∂K _T /∂T) _P , and (∂V _P /∂T) _P , where V _P is the compressional wave velocity, have smaller magnitudes than all other minerals of importance in Earth’s mantle, thus, confirming predictions from systematics studies. We find several dimensionless quantities for this monoclinic mineral have normal values compared to other mantle minerals. Further, αK _T (α is the volume coefficient of thermal expansion) for diopside is approximately independent of both T and volume V at elevated temperature, so its equation of state is accurately expressed in simplified form.     

New Approach for Estimation of Static and Seismic Active Earth Pressure

To estimate static and seismic active earth pressure (P_ad) on a rigid retaining wall, numerical analyses using different step sizes have been carried out in this paper, based on the modified Culmann line method by considering Coulomb’s planar rupture surface. Equivalent pseudo-static seismic forces are considered in the analysis. A new concept of modified unit weight by considering ground surcharge is introduced under static and seismic conditions. By numerical analysis, area of soil (A) has been estimated to obtain the ratio of A/A₀ where A₀ is θh², θ is the angle between retaining structure and ground surface and h is the vertical height of the wall. This ratio remains constant for a particular type of soil and has been used to estimate the maximum active earth pressure using force diagram. Results are provided in tabular form for easy calculation of the coefficient of static and seismic active earth pressure. Present results by considering the new technique, compares well with the results obtained by earlier researchers.     

P–V–T equations of state of MgO and thermodynamics

A simplest equation within the framework of the Mie-Grüneisen–Einstein approach is considered. Pressure estimation values are presented that are derived by conventional arithmetic and algebraic calculations as a function of temperature and volume. The equation under consideration complies with the Mie-Grüneisen–Debye model at high temperature. Different versions of an equation of state (EoS) of MgO proposed by Speziale et al. (J Geophys Res 106B:515–528, 2001) as a pressure standard at high temperatures are subject to analyses. In the literature, at least four versions of Speziale et al. EoS of MgO are discussed; the discrepancy between them reaching a few GPa at T > 2,000 K and P > 100 GPa. Our analyses of these equations suggest that the volume dependence of the Debye temperature is accepted arbitrarily and does not agree with the definition of the Grüneisen parameter, γ = −(∂lnΘ/∂lnV) _T . Pressure as a function of temperature and volume in the Mie-Grüneisen–Einstein approach or the Gao pressure calculator can be used to estimate true pressure at compression x = V/V ₀ < 1 with the Speziale et al. EoS of MgO.     

Pressure–volume–temperature equations of state: a comparative study based on numerical simulations

 The P–V–T equation of state (EoS) models of Birch–Murnaghan, Vinet and Poirier–Tarantola have been compared with one another and discussed in the light of their ability to reproduce thermoelastic functions and parameters by means of fitting to pressure–volume–temperature data artificially generated for spinel, corundum and forsterite. Numerical simulations relying upon semi-empirical potentials, lattice dynamics and the quasiharmonic approximation have been used to generate P–V–T data. The results obtained indicate that all the P–V–T EoSs tested predict bulk modulus at ambient conditions with errors confined, at worst, within a few percent, and reproduce correctly its dependence on temperature. The derivatives of the bulk modulus versus P and PT are less satisfactorily modelled. The bulk thermal expansion is determined by EoSs within a few percent error, but the deviations increase significantly if the approximation of linear dependence of EoS on temperature is used (linearised thermal pressure model). Received: 30 January 2001 / Accepted: 16 June 2001     

Locating a point on a spherical surface relative to a spherical polygon of arbitrary shape

An algorithm for determining if any given point,P, on the surface of a sphere is located inside, outside, or along the border of an arbitrary spherical polygon,S, is described. The polygon is described by specifying coordinates of its vertices, and coordinates of some pointX which is known to lie withinS. The algorithm is based on the principle that an arc joiningX andP will cross the border ofS an odd number of times ifP lies outsideS, and an even number of times ifP lies withinS. The algorithm has been implemented as a set of FORTRAN subroutines, and a listing is provided. The algorithm and subroutine package can be used with spherical polygons containing holes, or with composited spherical polygons.     

Computer modelling of a pressure induced phase change in clinoenstatite pyroxenes

 Using lattice dynamic modelling of pure MgSiO₃ clinopyroxenes, we have be able to simulate the properties of both the low-clino (P2₁/c) and a high-density-clino (C2/c) phases and our results are comparable with the high pressure (HP) X-ray study of these phases (Angel et al. 1992). The transition between the two phases is predicted to occur at 6GPa. The volume variation with pressure for both phases is described by a third-order Birch-Murnaghan equation of state with the parameters V _{0  low}=31.122 cm³·mol⁻¹, K _{T0  low}= 107.42 GPa, K′ _{T0  low}=5.96, V _{0  high}=30.142 cm³·mol^–1, K _{T0  high}102.54 GPa and K′ _{T0  high}=8.21. The change in entropy between the two modelled phases at 6GPa is ΔS _{6 Gpa}=−1.335 J·mol⁻¹·K⁻¹ and the equivalent change in volume is ΔV _{6 GPa}=−0.92 cm³·mol⁻¹, from which the gradient of the phase boundary δP/δT is 0.0014 GPa·K⁻¹. The variation of the bulk modulus with pressure was also determined from the modelled elastic constants and compares very well with the EOS data. The reported Lehmann discontinuity, ∼220 km depth and pressure of 7.11Gpa, has an increase in the seismic compressional wave velocity, v _p , of 7.14% using the data given for PREM (Anderson 1989). At a pressure of 7GPa any phase transition of MgSiO₃ pyroxene would be between ortho (Pbca) and high-clino. We find the value of v _p at 7GPa, for modelled orthoenstatite (Pbca), is 8.41 km·sec⁻¹ and that for the modelled high-clino phase at 7GPa is 8.93 km·sec⁻¹, giving a dv _p /v _p of 6.18%. Received: July 26, 1996 / Revised, accepted: September 27, 1996     

Crystal structure and compression of an iron-bearing Phase A to 33 GPa

Fe-bearing dense hydrous magnesium silicate Phase A, Mg_6.85Fe_0.14Si_2.00O₈(OH)₆ has been studied by single-crystal X-ray diffraction at ambient conditions and by high-pressure powder diffraction using synchrotron radiation to 33 GPa. Unit cell parameters at room temperature and pressure from single crystal diffraction are a=7.8678 (4) Å, c=9.5771 (5) Å, and V=513.43 (4) ų. Fitting of the P–V data to a third-order Birch-Murnaghan isothermal equation of state yields V ₀=512.3 (3) ų, K _T,0=102.9 (28) GPa and K′=6.4 (3). Compression is strongly anisotropic with the a-axes, which lie in the plane of the distorted close-packed layers, approximately 26% more compressible than the c-axis, which is normal to the plane. Structure refinement from single-crystal X-ray intensity data reveals expansion of the structure with Fe substitution, mainly by expansion of M-site octahedra. The short Si2–O6 distance becomes nearly 1% shorter with ~2% Fe substitution for Mg, possibly providing additional rigidity in the c-direction over the Mg end member. K _T obtained for the Fe-bearing sample is ~5.5% greater than reported previously for Fe-free Phase A, despite the larger unit cell volume. This study represents a direct comparison of structure and K _T–ρ relations between two compositions of a F-free dense hydrous magnesium silicate (DHMS) phase, and may help to characterize the effect of Fe substitution on the properties of other DHMS phases from studies of the Fe-free end-members.     

Synthetic light curves of close binaries in a cool-disk model. The “hot line” and “hot spot” as visual indicators of interaction between the flow and disk

We present a “combined” model taking into account visual manifestations of the interaction between the gas flow and the accretion disk in a close binary system in the form of a “hot line” and a “hot spot.” The binary consists of a red dwarf that fills its Roche lobe and a compact spherical star (a white dwarf or neutron star) surrounded with a thick ellipsoidal accretion disk of a complex shape. The disk thickness is not large near the compact star but increases according to a parabolic law towards its outer edge. The oblique collision of the gaseous flow with matter of the cool, rotating disk, whose outer edge has a temperature <10 000 K, creates an extended region of enhanced energy release. In the combined model, this region is represented with a hot line that coincides with the optically opaque part of the flow and is located outside the disk, together with a hot spot at the outer surface of the disk, on the leeward side of the flow. The synthetic light curves for the combinedmodel and a hot-line model demonstrate that both models are able to fairly accurately reproduce the shapes of both classical and atypical light curves of cataclysmic variables in quiescence. Our determination of the parameters of the cataclysmic variable OY Car from an analysis of its light curves using the two models shows that the basic characteristics of the close binary, such as the component mass ratio q = M ₁/M ₂, orbital inclination i, effective temperatures of the red dwarf (T ₂) and white dwarf (T ₁), and orientation of the disk α _e , remain the same within the errors. The parameters describing the size of the slightly elliptical disk and the radiation flux from the disk differ by several percent (∼ 2–8%). A more significant difference is detected in the parameters of the hot line, due to the different shape and alignment of the flow and the presence of an additional radiation source—the hot spot—on the disk.     

Qualitative and Quantitative Characteristics of Modeled and Natural Oscillatory Zoning Patterns in Calcite

Four types of oscillatory zoning patterns (OZP) produced by a dynamic model are described qualitatively and quantitatively and displayed as simulated cathodoluminescence images. The behavior of the dynamic model was investigated in terms of the parameter θ, which is the ratio of diffusivities of Ca²⁺ and H₂CO₃, and in terms of the parameter γ, which is the product of θ and the ratio of concentrations of Ca²⁺ and H₂CO₃ far away from the crystal surface. Qualitatively, the dynamics of the model has been characterized by a stable focus, an unstable focus changing to a stable node or to a stable limit cycle, and by periodical behavior with constant amplitude. Quantitative characteristics, including amplitude of oscillations and duration of oscillations, change between the patterns. It is shown that the process of forming oscillatory zoning in calcite with conditions corresponding to periodical behavior with constant amplitude is very slow in comparison to other OZPs. The oscillatory zoning pattern in a natural calcite crystal is described qualitatively in terms of four general OZPs produced by the dynamic model.     

Low T heat capacity measurements and new entropy data for titanite (sphene): implications for thermobarometry of high-pressure rocks

The accepted standard state entropy of titanite (sphene) has been questioned in several recent studies, which suggested a revision from the literature value 129.3 ± 0.8 J/mol K to values in the range of 110–120 J/mol K. The heat capacity of titanite was therefore re-measured with a PPMS in the range 5 to 300 K and the standard entropy of titanite was calculated as 127.2 ± 0.2 J/mol K, much closer to the original data than the suggested revisions. Volume parameters for a modified Murgnahan equation of state: V _P,T  = V ₂₉₈° × [1 + a°(T − 298) − 20a°(T − 298)] × [1 – 4P/(K ₂₉₈ × (1 – 1.5 × 10⁻⁴ [T − 298]) + 4P)]^1/4 were fit to recent unit cell determinations at elevated pressures and temperatures, yielding the constants V ₂₉₈° = 5.568 J/bar, a° = 3.1 × 10⁻⁵ K⁻¹, and K = 1,100 kbar. The standard Gibbs free energy of formation of titanite, −2456.2 kJ/mol (∆H°_f = −2598.4 kJ/mol) was calculated from the new entropy and volume data combined with data from experimental reversals on the reaction, titanite + kyanite = anorthite + rutile. This value is 4–11 kJ/mol less negative than that obtained from experimental determinations of the enthalpy of formation, and it is slightly more negative than values given in internally consistent databases. The displacement of most calculated phase equilibria involving titanite is not large except for reactions with small ∆S. Re-calculated baric estimates for several metamorphic suites yield pressure differences on the order of 2 kbar in eclogites and 10 kbar for ultra-high pressure titanite-bearing assemblages.     

Solution of Kepler’s equation for rectilinear motion

The properties of the solution of the kinematic equation (Kepler’s equation) presented as a series in powers of a function of time are considered for the case of unperturbed, rectilinear elliptical and rectilinear hyperbolic motion. Kepler’s equation for unperturbed, rectilinear elliptical motion has the form E − sinE = z ³/6, where E is the eccentric anomaly determining the position in the orbit, z ³/6 is the mean anomaly, which is proportional to the time measured from an encounter, and its solution can be represented as a series in powers of z. It is established that the coefficients of the series are positive. The asymptotic for the coefficients in the region of convergence of the series is found, which covers the entire orbit. The series continues to converge over the entire boundary of the circle of convergence. The kinematic equation for unperturbed, rectilinear hyperbolic motion has the form sinhH − H = ζ ³/6, where, as before, ζ ³ is proportional to time. The substitution E = iH, z = iζ reduces one equation to the other. The series for the solution in the hyperbolic case differs from the series for the elliptical solution only in its alternating-sign coefficients. However, the region of convergence covers only part of the orbit in the hyperbolic case.