Gravitational Hall Effect and Gravitomagnetic Dynamo

In the linear approximation of the gravitational field and using the gravitational quantum Hall effect, the gravitational conductivity is defined and the gravitomagnetic analogous of the kinematic dynamo is formulated.     

Some space gravity formulas and the dimensions and the mass of the Earth

Summary Adopting thePizzetti-Somigliana method and using elliptic integrals we have obtained closed formulas for the space gravity field in which one of the equipotential surfaces is a triaxial ellipsoid. The same formulas are also obtained in first approximation of the equatorial flattening avoiding the use of the elliptic integrals. Using data from satellites and Earth gravity data the gravitational and geometric bulge of the Earth's equator are computed. On the basis of these results and on the basis of recent gravity data taken around the equator between the longitudes 50° to 100° E, 155° to 180° E, and 145° to 180° W, we question the advantage of using a triaxial gravity formula and a triaxial ellipsoid in geodesy. Closed formulas for the space field in which a biaxial ellipsoid is an equipotential surface are also derived in polar coordinates and its parameters are specialized to give the international gravity formula values on the international ellipsoid. The possibility to compute the Earth's dimensions from the present Earth gravity data is the discussed and the value ofMG=(3.98603×10²⁰ cm³ sec^–2) (M mass of the Earth,G gravitational constant) is computed. The agreement of this value with others computed from the mean distance Earth-Moon is discussed. The Legendre polinomials series expansion of the gravitational potential is also added. In this series the coefficients of the polinomials are closed formulas in terms of the flattening andMG.Publication Number 327, and Istituto di Geodesia e Geofisica of Università di Trieste.     

Approaches indG/G-determinations

Summary Whereas observations of planets and solar satellites are more promising for detecting possible variations of Newtonian gravitational constant,G, with time the application of additional methods which are substantially different might be useful. It is discussed to what extent the tidal problems can be solved in connection with lunar orbit tracking for detectingG; further a relativeG-experiment at the earth surface is outlined.     

2017年精河MS6.6地震前重力变化特征分析

利用新疆地区北疆测网2015～2017年观测的共4期流动重力资料,分析研究区半年、1年尺度重力变化特征,并采用小波分解方法,对2017年8月9日精河M_S6.6地震前重力变化异常进行分解,分离了不同深度的重力异常。结果表明,震中附近的精河地区从2015年开始一直处于重力负值变化,同时,震中周围的重力正值变化集中区逐步向震中地区迁移,地震发生前震中以南60km处出现重力变化零值线,并且零值线两侧的重力累积变化量达到70μGal。小波分解后,4阶细节图较好地反映了精河地震前的重力异常,其与地震孕育发生有较好的对应性。     

RAPID COMPUTATION OF THE GRAVITATION ATTRACTION OF TOPOGRAPHY ON A SPHERICAL EARTH*

A brief review of the existing methods of gravity reduction is given and a new method suitable for use on high speed digital computers is described. The method is based on the formula for the gravitational attraction of a frustum of a cone. The topographic contours are represented by polygons and the x and y coordinates of corners of the polygons constitute the input to the computer. The vertical component of the gravitational attraction is calculated by evaluating the cone formula for a number of vertical sections of the topography. Each vertical section is simplified by adopting a procedure of grouping and averaging for the distant points of the section. The effect of the earth's sphericity is taken into account by lowering the distant points of the sections by amounts determined by the curvature. The computations include the area close to the point at which the attraction is required and may be limited to an area defined by a circle centered at this point. The method is therefore compatible with the conventional zone chart methods. As an illustration of the method the gravitational attraction of Caryn Seamount in the Atlantic Ocean is computed. The total Bouguer correction and the Terrain correction are also computed for an area in northwestern South America and comparisons are made with hand computations by a zone chart method. As an example, for work at sea, the Bouguer corrections for an area near the Island of Mauritius in the Indian Ocean are computed and the effects of sphericity and three-dimensionality are calculated. The gravitational attraction of two-dimensional bodies can be computed in a very similar manner. The attraction of the Puerto Rico Trench model is computed and the results are compared with other methods. The effects of sphericity and assumptions involved in extending the models to infinity are discussed.     

The gravitational field of topographic-isostatic masses and the hypothesis of mass condensation II-the topographic-isostatic geoid

In Part I we focussed on a convergent representation of the gravitational potential generated bytopographic masses on top of the equipotential surface atMean Sea Level, thegeoid, and by those masses which compensate topography. Topographic masses have also been condensated, namely represented by a single layer. Part II extends the computation of the gravitational field of topographic-isostatic masses by a detailed analysis of itsforce field in terms ofvector-spherical harmonic functions. In addition, the discontinuous mass-condensated topographic gravitational force vector (head force) is given. Once we identify theMoho discontinuity asone interface of isostatically compensated topographical masses, we have computed the topographic potential and the gravitational potential which is generated by isostatically compensated masses atMean Sea Level, the geoid, and illustrated by various figures of geoidal undulations. In comparison to a data oriented global geoid computation ofJ. Engels (1991) the conclusion can be made that the assumption of aconstant crustal mass density, the basic condition for isostatic modeling, does not apply. Insteaddensity variations in the crust, e.g. betweenoceanic and continental crust densities, have to be introduced in order to match the global real geoid and its topographic-isostatic model. The performed analysis documents that thestandard isostatic models based upon aconstant crustal density areunreal.     

The rotational and gravitational signature of the December 26, 2004 Sumatran earthquake

Besides generating seismic waves, which eventually dissipate, an earthquake also generates a static displacement field everywhere within the Earth. This global displacement field rearranges the Earth’s mass thereby causing the Earth’s rotation and gravitational field to change. The size of this change depends upon the magnitude, focal mechanism, and location of the earthquake. The Sumatran earthquake of December 26, 2004 is the largest earthquake to have occurred since the 1960 Chilean earthquake. Using a spherical, layered Earth model, the coseismic effect of the Sumatran earthquake upon the Earth’s length-of-day, polar motion, and low-degree harmonic coefficients of the gravitational field are computed. Using a model of the earthquake source that is composed of five subevents having a total moment-magnitude M _w of 9.3, it is found that this earthquake should have caused the length-of-day to decrease by 6.8 microseconds, the position of the Earth’s generalized figure axis to shift 2.32 milliarcseconds towards 127° E longitude, the Earth’s oblateness J ₂ to decrease by 2.37 × 10⁻¹¹ and the Earth’s pear-shapedness J ₃ to decrease by 0.63 × 10⁻¹¹. The predicted change in the length-of-day, position of the generalized figure axis, and J ₃ are probably not detectable by current measurement systems. But the predicted change in oblateness is perhaps detectable if other effects, such as those of the atmosphere, oceans, and continental water storage, can be adequately removed from the observations.     

APPROXIMATION OF THE GRAVITATIONAL ATTRACTION OF GEOLOGICAL BODIES*

The gravitational attraction produced by a geological body of irregular shape can be easily determined by dividing it into cubes of small size. The exact expression of the effect of a cube is very complicated, but it can be calculated by using an electronic computer. 4851 values of the gravitational attraction were determined for different positions of a cube with the side of 2l and the center in M(x₀, y₀, z₀), for x₀∈[0;20], y₀∈[0;20] and z₀∈[0;10]. Using these values, templates in double logarithmic representation were drawn, with x₀ and z₀ as parameters and y₀ as abscissa, and with x₀ and y₀ as parameters and z₀ in abscissa; this double set of templates permits a good interpolation for all cubes in the considered domain of M. The use of templates was tested to approximate the effect produced by a theoretical model of spherical shape and in a real case of a three-dimensional salt body of known shape based on a large number of boreholes. In both cases very good results were obtained.     

Internal waves in a rotating stratified fluid in an arbitrary gravitational field

Abstract

Small amplitude oscillations of a uniformly rotating, density stratified, Boussinesq, non-dissipative fluid are examined. A mathematical model is constructed to describe timedependent motions which are small deviations from an initial state that is motionless with respect to the rotating frame of reference. The basic stable density distribution is allowed to be an arbitrary prescribed function of the gravitational potential. The problem is considered for a wide class of gravitational fields. General properties of the eigenvalues and eigenfunctions of square integrable oscillations are demonstrated, and a bound is obtained for the magnitude of the frequencies. The modal solutions are classified as to type. The eigenfunctions for the pressure field are shown to satisfy a second-order partial differential equation of mixed type, and the equation is obtained for the critical surfaces which delineate the elliptic and hyperbolic regions. The nature of the problem is examined in detail for certain specific gravitational fields, e.g., a radially symmetric field. Where appropriate, results are compared with those of other investigations of waves in a rotating fluid of spherical configuration and the novel aspects of the present treatment are emphasized. Explicit modal solutions are obtained in the specific example of a fluid contained in a rigid cylinder, stratified in the presence of vertical gravity, with the buoyancy frequency N being an arbitrary prescribed function of depth.     

Quantification of peatland water storage capacity using the water table fluctuation method

Peat specific yield (S_Y) is an important parameter involved in many peatland hydrological functions such as flood attenuation, baseflow contribution to rivers, and maintaining groundwater levels in surficial aquifers. However, general knowledge on peatland water storage capacity is still very limited, due in part to the technical difficulties related to in situ measurements. The objectives of this study were to quantify vertical S_Y variations of water tables in peatlands using the water table fluctuation (WTF) method and to better understand the factors controlling peatland water storage capacity. The method was tested in five ombrotrophic peatlands located in the St. Lawrence Lowlands (southern Québec, Canada). In each peatland, water table wells were installed at three locations (up‐gradient, mid‐gradient, and down‐gradient). Near each well, a 1‐m long peat core (8 cm × 8 cm) was sampled, and subsamples were used to determine S_Y with standard gravitational drainage method. A larger peat sample (25 cm × 60 cm × 40 cm) was also collected in one peatland to estimate S_Y using a laboratory drainage method. In all sites, the mean water table depth ranged from 9 to 49 cm below the peat surface, with annual fluctuations varying between 15 and 29 cm for all locations. The WTF method produced similar results to the gravitational drainage experiments, with values ranging between 0.13 and 0.99 for the WTF method and between 0.01 and 0.95 for the gravitational drainage experiments. S_Y was found to rapidly decrease with depth within 20 cm, independently of the within‐site location and the mean annual water table depth. Dominant factors explaining S_Y variations were identified using analysis of variance. The most important factor was peatland site, followed by peat depth and seasonality. Variations in storage capacity considering site and seasonality followed regional effective growing degree days and evapotranspiration patterns. This work provides new data on spatial variations of peatland water storage capacity using an easily implemented method that requires only water table measurements and precipitation data.     

Determination of Q S, spreading factor and S-wave acceleration by inversion of seismic intensity data

In this paper, on the basis of intensity data from 85 earthquakes occurred in China, the method of resolving overdetermined equations by using the damped least squares method is applied to inversing for the mean value of 1 s. S wave quality factor, the spreading factor and the S-wave acceleration at the foci of earthquakes in 7 regions of China. The relation between S-wave acceleration at the foci of earthquakes and magnitude is discussed. As an example, 2-DQ _S distribution in Taiwan Province is obtained by inversion. It is found that there exists some corresponding relation between this distribution and Bouguer gravitational anomaly and seismicity. Preliminary analysis and discussion on results of the inversion, and estimation of error inQ _S are made. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,13, 202–211, 1991. The English version of this paper is improved by Professor Yushou Xie.     

Changes in the thermo-haline structure of the Dead Sea: 1979–1984

From 1979 to 1984, the overall water balance of the Dead Sea was characterized by a water deficit. However, an excess of freshwater inflow during the 1979/80 rainy season resulted in a 3-year-long meromictic phase. This was followed by three consecutive overturns of the water column in December 1982, 1983 and 1984. The buildup and dissipation of the seasonal thermocline and halocline is followed throughout this period which covers a wide range of water balance situations. The gravitational stabilities of the summer pycnoclines, measured in terms of N² 3×10⁻² s⁻², are at least one order of magnitude greater than the values reported in freshwater lakes and oceans. The contributions of temperature and salinity to N² and to the integrated stability W are examined separately, and their interdependence is pointed out. Two irreversible effects in the evolution of the properties of the water masses are identified: (1) a monotonic increase in the density of the deep waters; and (2) a monotonic shift of the NaCl saturation curve towards higher salinities.     

Turning surface behaviour for internal waves subject to general gravitational fields

Abstract

It is shown that the turning surfaces associated with internal waves in a uniformly rotating, density stratified, Boussinesq fluid in the presence of an arbitrary gravitational field are regular points for the governing eigenvalue differential equation. The results are illustrated for two particular examples that have geophysical and astrophysical significance, namely radially directed spherical gravity, and the gravitational field in a rapidly rotating cylinder.     

Hydrogeochemical balance of forest umbrisol profiles (‘Sierra de Gata’, central western Spain)

Three techniques for obtaining soil water solutions (gravitational and matrical waters extracted using both in situ tension lysimeters and in vitro pressure chambers) and their later chemical analysis were performed in order to know the evolution of the soil‐solution composition when water moves down through the soil, from the Ah soil horizon to the BwC‐ or C‐horizons of forest soils located in western Spain. Additionally, ion concentrations and water volumes of input waters to soil (canopy washout) and exported waters (drainage solutions from C‐horizons) were determined to establish the net balance of solutes in order to determine the rates of leaching or retention of ions. A generalized process of sorption or retention of most components (even Cl^?) was observed, from the soil surface to the C‐horizon, in both gravitational and matrical waters, with H₄SiO₄, Mn²⁺, Na⁺, and SO₄^2? being the net exported components from the soil through the groundwater. These results enhance the role of the recycling effect in these forest soils. The net percentages of elements retained in these forest soils, considering the inputs and the outputs balance, were 68% K⁺, 85% Ca²⁺, 58% Mg²⁺, 7% Al³⁺, 5% Fe³⁺, 34% Zn²⁺, 57% Cl^?, and 20% NO₃^?, and about 75% of dissolved organic carbon was mineralized. Copyright © 2007 John Wiley & Sons, Ltd.     

Determination of the Geopotential Scale Factor from TOPEX/POSEIDON Satellite Altimetry

The geopotential scale factor R _o = GM/W _o (the GM geocentric gravitational constant adopted) and/or geoidal potential Wo have been determined on the basis of the first year's (Oct 92 – Dec 93) ERS-1/TOPEX/POSEIDON altimeter data and of the POCM 4B sea surface topography model: R _o °=(6 363 672.58°±0.05) m, W _o °=(62 636 855.8°±0.05)m ² s ^–2 . The 2°–°3 cm uncertainty in the altimeter calibration limits the actual accuracy of the solution. Monitoring dW _o /dt has been projected.     

Effects of geomagnetic activity on the mesospheric electric fields

The results of three series of rocket measurements of mesospheric electric fields carried out under different geomagnetic conditions at polar and high middle latitudes are analysed. The measurements show a clear dependence of the vertical electric fields on geomagnetic activity at polar and high middle latitudes. The vertical electric fields in the lower mesosphere increase with the increase of geomagnetic indexes K_p and K_p. The simultaneous increase of the vertical electric field strength and ion conductivity was observed in the mesosphere during geomagnetic disturbances. This striking phenomenon was displayed most clearly during the solar proton events of October, 1989 accompanied by very strong geomagnetic storm (K_p = 8+). A possible mechanism of generation of the vertical electric fields in the mesosphere caused by gravitational sedimentation of charged aerosol particles is discussed. Simultaneous existence in the mesosphere of both the negative and positive multiply charged aerosol particles of different sizes is assumed for explanation of the observed V/m vertical electric fields and their behaviour under geomagnetically disturbed conditions.Paper Presented at the Second IAGA/ICMA (IAMAS) Workshop on Solar Activity Forcing of the Middle Atmosphere, Prague, August 1997     

Spherical Harmonic Analysis of Gravitational Curvatures and Its Implications for Future Satellite Missions

In this study we assume that a gravitational curvature tensor, i.e. a tensor of third-order directional derivatives of the Earth’s gravitational potential, is observable at satellite altitudes. Such a tensor is composed of ten different components, i.e. gravitational curvatures, which may be combined into vertical–vertical–vertical, vertical–vertical–horizontal, vertical–horizontal–horizontal and horizontal–horizontal-horizontal gravitational curvatures. Firstly, we study spectral properties of the gravitational curvatures. Secondly, we derive new quadrature formulas for the spherical harmonic analysis of the four gravitational curvatures and provide their corresponding analytical error models. Thirdly, requirements for an instrument that would eventually observe gravitational curvatures by differential accelerometry are investigated. The results reveal that measuring third-order directional derivatives of the gravitational potential imposes very high requirements on the accuracy of deployed accelerometers which are beyond the limits of currently available sensors. For example, for orbital parameters and performance similar to those of the GOCE mission, observing third-order directional derivatives requires accelerometers with the noise level of \({\sim}10^{-17}\,\hbox {m}\,\hbox {s}^{-2}\) Hz\(^{-1/2}\).     

Planetary structures in general relativity

Summary For a static single-zone planet assuming that 1) the pressure and density are connected by the equationP =K _Q ^1+1/n–s, 2) a complete spherical symmetry is preserved and the system is in hydrostatic equilibrium; in section II, the expressions for the field equations have been obtained in suitable dimensionless forms. In section III the solution of the field equations forn=0 (which represents a homogeneous liquid) has been given in explicit form; for other prescribed value ofn=2/5 it has been pointed out that solutions must be performed by numerical integrations. Expressions for the mass-radius relation, the ratio of central to average density, the total energy, the proper energy and gravitational potential energy, which give some informations about the internal structure of the planet, have also been given in this section. Section IV discusses the velocity of sound at the centre of the planet. A few concluding remarks regarding the structure of the planet have been given in section V.     

On The Determination Of The Earth's Model – The Mean Equipotential Surface

The sea surface cannot be used as reference for Major Vertical Datum definition because its deviations from the ideal equipotential surface are very large compared to rms in the observed quantities. The quasigeoid is not quite suitable as the surface representing the most accurate Earth's model without some additional conditions, because it depends on the reference field. The normal Earth's model represented by the rotational level ellipsoid can be defined by the geocentric gravitational constant, the difference in the principal Earth's inertia moments, by the angular velocity of the Earth's rotation and by the semimajor axis or by the potential (U ₀ ) on the surface of the level ellipsoid. After determining the geopotential at the gauge stations defining Vertical Datums, gravity anomalies and heights should be transformed into the unique vertical system (Major Vertical Datum). This makes it possible to apply Brovar's (1995) idea of determining the reference ellipsoid by minimizing the integral, introduced by Riemann as the Dirichlet principle, to reach a minimum rms anomalous gravity field. Since the semimajor axis depends on tidal effects, potential U ₀ should be adopted as the fourth primary fundamental geodetic constant. The equipotential surface, the actual geopotential of which is equal to U ₀ , can be adopted as reference for realizing the Major Vertical Datum.