Vertical motion and diabatic heating over the Indian monsoon region during the Bay of Bengal depression of 5–8 July, 1979

The vertical velocity, , and the diabatic heating were computed at 800, 600, 400 and 200 mb surfaces using the Omega equation. The highest contribution to is from the diabatic heating produced by condensation associated with the precipitations appearing to be the main source of diabatic heating. The net radiative cooling and the thermal advection in the upper troposphere over the warm anticyclone result in diabatic cooling over the eastern part of the Bay of Bengal and adjoining northern and eastern regions.List of Symbols Used C _p Heat capacity at constant pressure - f Coriolis parameter - g Acceleration due to gravity - P Atmospheric pressure - Q Diabatic heating rate per unit mass - R Gas constant of air - S Static stability parameter - t Time - U, V Zonal and meridional wind components - Specific volume - Relative vorticity - Absolute vorticity - Potential temperature - Geopotential - Vertical velocity (dP/dt) - ₁ Adiabatic vertical velocity - ₂ Vertical velocity due to certain forcing - ₃ Diabatic vertical velocity - Isobaric gradient operator - ² Laplacian operator - J(A, B) Jacobian operator     

Some laboratory model experiments on the mechanism of two-dimensional momentum transfer in the atmosphere

Summary Investigating the jetstream formation in the upper troposphere, it becomes evident that two-dimensional transfer processes are, to a high degree, responsible for the accumulation of kinetic energy and momentum in those upper tropospheric regions. To shed light on these processes a series of barotropic model experiments has been carried out using a special dynamic procedure for vortex generation. In a large, rotating, cylindrical vessel (filled with water, rotation rate ₀) three smaller cylinders have been fitted eccentrically. Their rotation relative to the vessel (rotation rate _i ,i=1, 2, 3) produces a sharply limited narrow region around each of them, where a nearly two-dimensional relative motion can be observed. Beyond this region — we call it friction zone — there is no motion. Now if we impose some perturbation on this steady mean flow (e.g. by setting _i = 0) the zonal current breaks down into a definite number of vortices which show a random distribution. The dynamic behaviour of these nearly two-dimensional vortices can then be studied.The main results are: I) After the sudden breakdown of the friction zones the most intensive eddies, which are cyclonic, drift more or less directly towards the centre to be organized there into a larger vortex. This development is strongly influenced by the -effect and the ratio = _i /₀, which controls the supply of rotational energy. II) If there exists a steady zonal basic current, the vortices will be more and more deformed when approaching and penetrating the zonal flow until they are wholly incorporated.     

Consequences of induced polarization in magneto-telluric interpretation

Summary Both in-situ and laboratory studies indicate that naturally occurring earth materials are electrically dispersive. A review of both historical and recent developments in the field of dispersion is presented, including induced polarization (IP) as used in mineral prospecting. Computer-simulated magnetotelluric (MT) sounding curves are presented for nondispersive, semi-dispersive and dispersive, electrical structures. Admittivities of the form =A +A log (/₀)and other frequency dependent forms are used in the calculations. The results indicate that interpretation of electromagnetic field data, in terms of nondispersive layered structures, can lead to significant errors in the estimation of subsurface structure. Thus, there may be an important limitation in the use of variable frequency or transient data in the currently popular inversion schemes.     

A new dissipation model based on memory mechanism

Summary The model of dissipation based on memory introduced by Caputo is generalized and checked with experimental dissipation curves of various materials.List of symbols unidimensional stress - unidimensional strain - Q ^–1 specific dissipation function - c(t) creep compliance - m(t) relaxation modulus - c ₀ instantaneous compliance - m equilibrium modulus - (t) creep function - relaxation function - () spectral distribution of retardation times - spectral distribution of relaxation times - c ^*() complex compliance - m ^*() complex modulus - tang loss-tangent     

Molecular refraction in the Earth's mantle

The Drude law (molecular refraction) for the temperature radiation in a monoatomic model of the Earth's mantle is derived. The considerations are based on the Lorentz electron theory of solids. The characteristic frequency (or eigenfrequency) of independent electron oscillators (in energy units, ) is identified with the band gapE _G of a solid. The only assumption is that solid material related to the Earth's mantle has the mean atomic weight A21 g/mole, and its energy gap (E _G) is about 9 eV. In this case the value of molecular refraction (in cm³/g) is (n ²–1)/=0.5160.52, where andn are the density and the refractive index at wavelength _D=0.5893 m (sodium light), respectively. The average molecular refraction of important silicate and oxide minerals with A21, obtained byAnderson andSchreiber (1965) from laboratory data, is , where denotes the mean arithmetic value calculated from three principal refractive indices of crystal. For the rock-forming minerals with 19A<24 g/mole the new relation was found byAnderson (1975).     

Anisotropic magnetic susceptibility of rocks under mechanical stresses

Summary The magnetic susceptibility of a rock under a uniaxial compression () decreases along the axis of compression and increases along the direction perpendicular to the axis, with an increase of . Thus, the magnetic susceptibility of a compressed rock becomes anisotropic.The decrease of longitudinal susceptibility,K (), and the increase of transverse susceptibility,K (), are theoretically derived from a model of rock which assumes the uniaxial anisotropy and the isotropic magnetostriction of magnetic minerals in rocks and a random orientation of the minerals. Results show thatK () decreases toward zero whereasK () increases and approaches a finite asymptotic value with an increase of , and –(/)K () is twice as large as /K () for small values of . These results are in good agreement with experimental data.

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Zusammenfassung Die magnetische Suszeptibilität eines Steines unter zunehmender uniachsigen Druckspannung () nimmt ab längs der Achse der Druckspannung und nimmt zu längs der Richtung senkrecht der Achse. Somit wird die magnetische Suszeptibilität des gedrückten Steines anisotrop.Die Abnahme der longitudinalen Suszeptibilität,K (), und die Zunahme der transversalen Suszeptibilität,K (), werden theoretisch von einem Modell eines Steines hergeleitet, das die uniachsige Anisotropie, die isotrope Magnetostriktion, und eine nichtbevorzugte Orientierung der magnetischen Minerals im Stein annimmt. Die Ergebnisse zeigen, dass mit einer Zunahme des ,K () gegen Null abnimmt, währendK () zunimmt und sich einem begrenzten asymtotitschen Wert nähert und, dass für kleine Werte des , –(/)K () zweimal so gross wie /K () ist. Diese Ergebnisse stimmen gut mit den Versuchangaben überein.

     

Scattering attenuation and the fractal geometry of fracture systems

Scattering of seismic waves can be shown to have a frequency dependenceQ ^{–1 3–v} if scattering is produced by arrays of inhomogeneities with a 3D power spectrumW _3D(k) k ^–v. In the earth's crust and upper mantle the total attenuation is often dominated by scattering rather than intrinsic absorption, and is found to be frequency dependent according toQ ^–1 , where –1<–0.5. IfD ₁ is the fractal dimension of the surface of the 3D inhomogeneities measured on a 2D section, then this corresponds respectively to 1.5<D ₁1.75, since it can be shown that =2(D ₁–2). Laboratory results show that such a distribution of inhomogeneities, if due to microcracking, can be produced only at low stress intensities and slow crack velocities controlled by stress corrosion reactions. Thus it is likely that the earth's brittle crust is pervaded by tensile microcracks, at least partially filled by a chemically active fluid, and preferentially aligned parallel to the maximum principal compressive stress. The possibility of stress corrosion implies that microcracks may grow under conditions which are very sensitive to pre-existing heterogeneities in material constants, and hence it may be difficult in practice to separate the relative contribution of crack-induced heterogeneity from more permanent geological heterogeneities.By constrast, shear faults formed by dynamic rupture at critical stress intensities produceD ₁=1, consistent with a dynamic rupture criterion for a power law distribution of fault lengths with negative exponentD. The results presented here suggest empirically thatD ₁-1/2(D+1), thereby providing the basis for a possible framework to unify the interpretation of temporal variations in seismicb-value (b-D/2) and the frequency dependence of scattering attenuation ().This is PRIS contribution 046.     

Source spectra and RMS acceleration of Mexican subduction zone earthquakes

Strong motion (SM) data of six Mexican subduction zone earthquakes (6.4M _S8.1) recorded near the epicentral zone are analyzed to estimate their far-field source acceleration spectra at higher frequencies (f0.3 Hz). Apart from the usual corrections such as geometrical spreading (1/R), average radiation pattern (0.6), free surface amplification (a factor of 2), and equal partitioning of the energy into two orthogonal horizontal components (a factor of 1/ ), the observed spectra are corrected for a frequency dependentQ(Q=100f), a site dependent filter (e ^–kf ), and amplification ofS waves near the surface (a factor of about 2 atf2Hz). We takeR as the average distance from the rupture area to the site. If we model the high frequency plateau (f1 Hz) of the source spectra, by a point source ^–2-model, and interpret them in terms of Brune's model we obtain between 50 and 100 bars for all earthquakes. The low-frequency broadband teleseismicP wave spectra, corrected witht ^*=1.0 s, agrees within a factor of two with SM source spectra near 1 Hz. The ^–2-model is inadequate to explain the observed source spectra in a broad frequency range; these resemble spectra given byGusev (1983) with some differences.SM source acceleration spectra require significant corrections to explain observed spectra and RMS acceleration (a_rms) (a) at farther coastal sites for extended sources due to directivity effect and (b) at inland sites (100R200 km) because of unaccounted path and site amplification and/or invalidity of body-wave approximation. The observed spectra and a_rms at these sites are significantly greater than the predicted values from the estimated source spectra.     

On the frequency modulation of VLF emissions

The VLF-wave frequency modulation efficiency as a function of magnetosphere plasma parameters under the weak pitch-angle diffusion regime is studied. The study is based on the VLF growth-rate modulation both in the magnetosphere equatorial plane and after integrating along the magnetic field line. It is shown that for the typical quiet magnetosphere plasma parameters the relative shift of the maximum intensity frequency _m/_m is approximately equal to relative disturbance of the magnetic field in the magnetosphere equatorial plane, but may exceed it when both electron temperature anisotropy and the parameter are small; here is the total-to-warm electron content ratio and is the electron parallel beta. It is also shown that relative shift of the maximum intensity frequency _m/_m after integrating along the field line is not less than 50% from its value at the equatorial plane, which allows the use of the equatorial-plasma-parameter data to estimate the VLF frequency modulation on the ground. The upper cut-off frequency modulation is considered as well. The calculated theoretical sonagrams show that this frequency modulation may be related to the non-dispersive and to the inverted-V structures of QP hiss.     

Die heterogenen sphäroidischen Gleichgewichtsfiguren und das Normalsphärold der Erde

Zusammenfassung Die Gleichgewichtsfiguren lassen sich gänzlich unabhängig vom Dichtegesetz durch die Eingeschaft charakterisieren, daß der Absolutbetrag des Formparametersf0 ein Minimum sein muß. Diese merkwürdige Eigenschaft liefert eine Gleichung zwischen der geometrischen Abplattung und den beidenStokesschen Konstanten und , mit deren Hilfe aus den ⁴ Lösungen desHelmertschen Gleichungssystems für eine bestimmte Masse die ³ Gleichgewichtsfiguren ausgesiebt werden können. Jede beliebige heterogene sphäroidische Gleichgewichtsfigur ist entweder durch die Masse und die Gestalt ihrer freien Oberfläche oder durch die Masse und drei physikalische Parameter gänzlich eindeurig bestimmt; sie hat ein streng individuelles Dichtegesttz. Aus der dreifach unendlichen Mannigfaltigkeit der Gleichgewichtsfiguren können linear Reihen herausgegriffen werden, indem man entweder zwei physikalische Parameter festhält oder indem man die Figuren aufsucht, welche eine gegebene Fläche zur gemeinsamen äußeren Niveaufläche besitzen oder die Reihe jener Gleichgewichtsfiguren, die sich aus der Schar der äußeren Niveauflähen einer gegebenen Gleichgewichtsfigur bilden läßt.Obwohl das HauptträgheitsmomentC keineStockessche Konstante ist, kann das durch ,W ₀ undC eindeutig definierte Normalsphäroid der Erde hypothesenfrei bestimmt werden, weil in der Reihe (,K) auch die Trägheitsmomente und damit die dynamische Abplattung konstant ist. Damit kann die empirisch bekannte dynamische Abplattung mittels des Rückganges auf die homogene Ausgangsfigur der Reihe (,K) durch die statische Abplattung ersetzt werden. Allerdings muß der Ableitung des Normalsphäroides an Stelle der primär unbekannten Werte für die Erdmasse und den Potentialwert des Geoides die Äquatorschwere und die Äquatorachse zugrunde gelegt werden.Abschließend werden noch die drei linearen Reihen (,W ₀), (,K) und (W ₀,K) diskutiert, welche sich im Normalsphäroid schneiden müssen. Auch kann in dem mit den Achsena, undh _m gebildeten Koordinatensystem die Hüllfläche der Gleichgewichtsfiguren angegeben werden; sie ist durch den Formparameterf=–3a²/2 gekennzeichnet.

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Summary The figures of equilibrium independant from the law of density in their interior can be characterized by the remarkable property that the absolute value of the «shape-parameter» must be a minimum. This gives an equation between the flattening and the twoStokes constant and , by which the ³ figures of equilibrium can be selected from the ⁴ solutions ofHelmert's equations for a given mass. Each inhomogeneous spheroidical figure of equilibrium is determined unequivocally by the mass and the shape of its free surface or by the mass and three physical parameters; the law of density is strictly individual. From the threefold infinite multiplicity of the figures of equilibrium linear series are to find out with two fixed physical parameters or you can compute a series of figures with a common level surface or the series built by all level surface of a given figure of equilibrium.Though the moments of inertia are notStokes constants the normal spheroid of earth unequivocally defined by ,W ₀ andC can be determined without hypotheses, because in the series (,K) also the moments of inertia and the mechanical ellipticity are constant. Therefore the empirically known mechanical ellipticity can be substituted by the static flattening returning to the homogenous figure in the beginning of the series. Of course the determination of the normal spheroid of earth demands the knowledge of gravity in equator and of the equator-axis instead of the unknown mass of earch series (,W ₀), (,K) and (K,W ₀), which intersect in the normal spheroid, are discussed. Also the boundary surface of the figures of equilibrium in the Cartesian system of the coordinatesa, andh _m is discussed; this surface is determined byf=–3a²/2.

     

Frequency dependence of crustalQ β in stable and tectonically active regions

Fundamental-mode Rayleigh wave attenuation data for stable and tectonically active regions of North America, South America, and India are inverted to obtain several frequency-independent and frequency-dependentQ models. Because of trade-offs between the effect of depth distribution and frequency-dependence ofQ on surface wave attenuation there are many diverse models which will satisfy the fundamental-mode data. Higher-mode data, such as 1-Hz Lg can, however, constrain the range of possible models, at least in the upper crust. By using synthetic Lg seismograms to compute expected Lg attenuation coefficients for various models we obtained frequency-dependentQ models for three stable and three tectonically active regions, after making assumptions concerning the nature of the variation ofQ with frequency.In stable regions, ifQ varies as , where is a constant, models in which =0.5, 0.5, and 0.75 satisfy fundamental-mode Rayleigh and 1-Hz Lg data for eastern North America, eastern South America, and the Indian Shield, respectively. IfQ is assumed to be independent of frequency (=0.0) for periods of 3 s and greater, and is allowed to increase from 0.0 at 3 s to a maximum value at 1 s, then that maximum value for is about 0.7, 0.6, and 0.9, respectively, for eastern North America, eastern South America, and the Indian Shield. TheQ models obtained under each of the above-mentioned two assumptions differ substantially from one another for each region, a result which indicates the importance of obtaining high-quality higher-mode attenuation data over a broad range of periods.Tectonically active regions require a much lower degree of frequency dependence to explain both observed fundamental-mode and observed Lg data. Optimum values of for western North America and western South America are 0.0 if is constant (Q is independent of frequency), but uncertainty in the Lg attenuation data allows to be as high as about 0.3 for western North America and 0.2 for western South America. In the Himalaya, the optimum value of is about 0.2, but it could range between 0.0 and 0.5. Frequency-independent models (=0.0) for these regions yield minimumQ values in the upper mantle of about 40, 70, and 40 for western North America, western South America, and the Himalaya, respectively.In order to be compatible with the frequency dependence ofQ observed in body-wave studies,Q in stable regions must be frequency-dependent to much greater depths than those which can be studied using the surface wave data available for this study, andQ in tectonically active regions must become frequency-dependent at upper mantle or lower crustal depths.On leave from the Department of Geophysics, Yunnan University, Kunming Yunnan, People's Republic of China     

On the Upper Bound of the Liquid Core Viscosity

By using Molodensky and Sasao (1995a) and Molodensky and Groten (1998) an approach of expansion in powers of the small parameter =(+)/ (where + and are the frequencies of nutational motion in space and in a mantle-fixed reference frame, respectively, is tidal frequency) a theory of diurnal Earth tides and nutation for a realistic model of the Earth with an inhomogeneous, viscous liquid core and an anelastic mantle is constructed. It is shown that our approach is self-consistent for semi-annual, annual, and principal nutational components (when ||1/180). By comparing the results of modern VLBI-nutational data and the results of our calculations, we have found the region of possible values of the parameters which describe the anelastic properties of the Earth's lower mantle and the viscosity of the liquid core. It is shown that modern VLBI-data are about six orders of magnitude more sensitive to the liquid core viscosity than modern seismic data or Earth free oscillations data.     

Structure and evolution of the terrestrial planets

Geo-scientific planetary research of the last 25 years has revealed the global structure and evolution of the terrestrial planets Moon, Mercury, Venus and Mars. The evolution of the terrestrial bodies involves a differentiation into heavy metallic cores, Fe-and Mg-rich silicate mantles and light Ca, Al-rich silicate crusts early in the history of the solar system. Magnetic measurements yield a weak dipole field for Mercury, a very weak field (and local anomalies) for the Moon and no measurable field for Venus and mars. Seismic studies of the Moon show a crust-mantle boundary at an average depth of 60 km for the front side, P- and S-wave velocities around 8 respectively 4.5 km s^–1 in the mantle and a considerable S-wave attenuation below a depth of 1000 km. Satellite gravity permits the study of lateral density variations in the lithosphere. Additional contributions come from photogeology, orbital particle, x-and -ray measurements, radar and petrology.The cratered surfaces of the smaller bodies Moon and Mercury have been mainly shaped by meteorite impacts followed by a period of volcanic flows into the impact basins until about 3×10⁹ yr before present. Mars in addition shows a more developed surface. Its northern half is dominated by subsidence and younger volcanic flows. It even shows a graben system (rift) in the equatorial region. Large channels and relics of permafrost attest the role of water for the erosional history. Venus, the most developed body except Earth, shows many indications of volcanism, grabens (rifts) and at least at northern latitudes collisional belts, i.e. mountain ranges, suggesting a limited plate tectonic process with a possible shallow subduction.List of Symbols and Abbreviations a=R _e mean equatorial radius (km) - A(r, t) heat production by radioactive elements (W m^–3) - A, B equatorial moments of inertia - b polar radius (km) - complex amplitude of bathymetry in the wave number (K) domain (m) - C polar moment of inertia - C _Fe moment of inertia of metallic core - C _Si moment of inertia of silicate mantle - C _p heat capacity at constant pressure (JK^–1 mole^–) - C _nm,J _nm,S _nm harmonic coefficients of degreen and orderm - C/(MR _e ² ) factor of moment of inertia - d distance (km) - d nondimensional radius of disc load of elastic bending model - D diameter of crater (km) - D flexural rigidity (dyn cm) - E Young modulus (dyn cm^–2) - E maximum strain energy - E energy loss during time interval t - f frequency (Hz) - f flattening - F magnetic field strength (Oe) (1 Oe=79.58A m^–1) - g acceleration or gravity (cms^–2) or (mGal) (1mGal=10^–3cms^–2) - mean acceleration - g _e equatorial surface gravity - complex amplitude of gravity anomaly in the wave number (K) domain - g free air gravity anomaly (FAA) - g Bouguer gravity anomaly - g _t gravity attraction of the topography - G gravitational constant,G=6.67×10^–11 m³kg^–1s^–2 - GM planetocentric gravitational constant - h relation of centrifugal acceleration (² R _e ) to surface acceleration (g _e ) at the equator - J magnetic flux density (magnetic field) (T) (1T=10⁹ nT=10⁹ =10⁴G (Gauss)) - J ₂ oblateness - J _nm seeC _nm - k ₍₀₎ (zero) pressure bulk modulus (Pa) (Pascal, 1 Pa=1 Nm^–2) - K wave number (km^–1) - K ^* thermal conductivity (Jm^–1s^–1K^–1) - L thickness of elastic lithosphere (km) - M mas of planet (kg) - M _Fe mass of metallic core - M _Si mass of silicate mantle - M(r) fractional mass of planet with fractional radiusr - m magnetic dipole moment (Am²) (1Am²=10³Gcm³) - m _b body wave magnitude - N crater frequency (km^–2) - N(D) cumulative number of cumulative frequency of craters with diameters D - P pressure (Pa) (1Pa=1Nm^–2=10^–5 bar) - P _z vertical (lithostatic) stress, see also _z (Pa) - P _n ^m (cos) Legendre polynomial - q surface load (dyn cm^–2) - Q seismic quality factor, 2E/E - Q _s ,Q _p seismic quality factor derived from seismic S-and P-waves - R=R ₀ mean radius of the planet (km) (2a+b)/3 - R _e =a mean equatorial radius of the planet - r distance from the center of the planet (fractional radius) - r _Fe radius of metallic core - S _nm seeC _nm - t time and age in a (years), d (days), h (hours), min (minutes), s (seconds) - T mean crustal thickness from Airy isostatic gravity models (km) - T temperature (°C or K) (0°C=273.15K) - T _m solidus temperature - T sideral period of rotation in d (days), h (hours), min (minutes), s (seconds), =2/T - U external potential field of gravity of a planet - V volume of planet - V _p ,V _s compressional (P), shear (S) wave velocity, respectively (kms^–1) - w deflection of lithosphere from elastic bending models (km) - z, Z depth (km) - z (K) admittance function (mGal m^–1) - thermal expansion (°C^–1) - viscosity (poise) (1 poise=1gcm^–1s^–1) - co-latitude (90°-) - longitude - Poisson ratio - density (g cm^–3) - mean density - ₀ zero pressure density - _m , _Si average density of silicate mantle (fluid interior) - average density of metallic core - _t , _top density of the topography - density difference between crustal and mantle material - electrical conductivity (^–1 m^–1) - _r , radial and azimuthal surface stress of axisymmetric load (Pa) - _z vertical (lithostatic) stress (seeP _z ) - _II second invariant of stress deviation tensor - latitude - angular velocity of a planet (=2/T) - ages in years (a), generally 0 years is present - B.P. before present - FAA Free Air Gravity Anomaly (see g - HFT High Frequency Teleseismic event - LTP Lunar Transient Phenomenon - LOS Line-Of-Sight - NRM Natural Remanent Magnetization Contribution No. 309, Institut für Geophysik der Universität, Kiel, F.R.G.     

A note on tidal current rotation

The rotational form of the vertically averaged equations of motion is applied to derive a formula, linear friction included, which establishes a direct connection between sense of rotation of tidal currents and features of tidal amphidromic systems. Two factors in the formula, called and , influence the sense of rotation of tidal currents; the factor involves the frequency of the tidal signal , the Coriolis parameter f, and the linear friction coefficient r. The sign of the cross-product of the logarithm of sea-surface elevation (), and phase () gradients determines whether the factor favors clockwise or anticlockwise sense of rotation. is a unit vector and is the angle between ln and . The limits ||0, ||0 and 0 lead to a clockwise sense of rotation in the Northern Hemisphere. 0 favors anticlockwise rotation in the Northern Hemisphere. Friction and low frequencies favor an anticlockwise sense of rotation. The theory works well in semi-enclosed regions like the North Sea. Although only linear friction and sea-surface elevation gradients were considered, there are ocean regions where the agreement between theory and observations is also good.Responsible Editor: Hans Burchard     

Magnitudes of detectable and localizable shocks in nw bohemia

Summary The effectiveness of recording seismic phenomena in the Kruné hory (Mts.) region in NW Bohemia by selected stations in the CSR, GDR and Poland has been estimated. Magnitude isolines of the weakest earthquakes, which can be localized and detected with an 0.9 probability, were calculated on the basis of the level of seismic disturbances at the individual stations and of the empirical dependence of the attenuation of seismic waves with distance.

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a a mum umauu uu u amu ¶rt; ana¶rt; uu uau mauu a mumuu , u a a uu n a m¶rt; mau u nuu auumu amau uu m amu u auma uuuu aum¶rt; a a mu, m mm 0.9 auuam u aum.

     

A note on upper limits for hypothetic variation in the Newtonian constant of gravitation

Summary It has been demonstrated on the basis of recent astronomical, satellite and LLR data that the variations in the Newtonian constant of gravitation, if any, do not exceed5××10 ^–15 cy^–1 of its relative value.

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¶rt;a amuu u nmu a¶rt;u u a auu naa, m auauuaumau nm, u u um, n¶rt;m5×10 ^–15 mmu ^–1 mum au.

     

On the rotational motion of deformable celestial bodies

¶rt;am au uu ¶rt; aam ¶rt;uu ¶rt;u ma; ¶rt;am am u ¶rt; a, ¶rt;a u¶rt; amua m uuu a; num n aaa auauu nm.     

Die Abplattungsfunktion einparametriger sphäroidischer Gleichgewichtsfiguren

Zusammenfassung Aus der schon früher bewiesenen Eindeutigkeit des Dichtegesetzes der sphäroidischen Gleichgewichtsfiguren ergibt sich die Möglichkeit, aus einer gegebenen Gleichgewichtsfigur durch «Entblätterung» eine unendliche Reihe neuer Gleichgewichtsfiguren derselben Rotationsgeschwindigkeit mit ständig abnehmender Größe und Masse zu bilden. Auf diese Weise kann man jede beliebige innere Niveaufläche der ursprünglichen Figur bloßlegen und aus der von ihr umschlossenen TeilmasseE, der Rotationsgeschwindigkeit , der Äquatorachsea und der Äquatorachsea _h des homogenen Ausgangsellipsoides der zugehörigen Figurenreihe (,C) die Abplattung berechnen und gewinnt so die Abplattungsfunktion im Innern einer Gleichgewichtsfigur gänzlich unabhängig von derClairautschen Differentialgleichung. Die Methode der Entblätterung ist auch auf höherparametrige Gleichgewichtsfiguren mit unstetigem Dichtegesetz anwendbar; sie liefert für dasWiechertsche Modell die Kernoberfläche in einer Tiefe von 3864.75 km und einen Dichtesprung von 17.84.

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Summary In a previous paper I have shown, that the law of density of the spheroidal figures of equilibrium is strictly individual. This makes it possible from a given figure of equilibrium by the so called method of «stripping of leaves» to gain an infinite series of new figures of equilibrium with permanently decreasing size and mass. In this manner any inner level surface of the given figure can be laid bare and its flattening can be calculated from the enclosed part of massE, the rotation velocity , the equator axisa and from the equator axisa _h of theMacLaurin ellipsoid in the beginning of the series (,C) belonging to the seeked figure. So the function of flattening in the interior of any equilibrium figure is gained totally independant fromClairaut's differential equation. The method of «stripping of leaves» also can be used for figures of equilibrium with more than one form-parameter and with discontinual density distribution. It yields forWiechert's model the surface of the core in a depth of 3864.75 km and a discontinuity of 17.84 in density.

     

Sealing law of far-field spectra based on observed parameters of the specific barrier model

A set of acceleration source spectra is constructed using the observed parameters of the specific barrier model of Papageorgiou and Aki. The spectra show a significant departure from the ²-model at the high frequency range. Specifically, the high frequency spectral amplitudes of seismic excitation are higher as compared to the level predicted by the ²-model. This is also supported by other observational evidence. The high frequency amplitudes of acceleration scale proportionally to the square root of the rupture areaS, to the rupture spreading velocityv, and to the local strain drop (/) (=strain drop in between barriers). The local strain drop in between barriers is not related in a simple way to the global strain drop, which is the strain drop estimated by assuming that it is uniform over the entire rupture area. Consequently, the similarity law does not apply. Using the source spectra which we constructed, we derive expressions for high frequency amplitudes of acceleration such asa _rms anda _max. Close to the fault both are independent of fault dimensions and scale as (/µ)(f)^1/2, while away from the fault plane they scale asW ^1/2(/µ)(f)^1/2, whereW is the width of the fault and f is the effective bandwidth of the spectra.     

Power radiated by a VLF loop antenna in the ionospheric plasma

Summary The radiation power a VLF loop antenna with an arbitrary orientation of the loop's plane relative to the direction of the external magnetic field is calculated and its portion, transferred to the electromagnetic part of the excited spectrum, is determined.

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am umaa m uu am c nu umau nmu uma n m¶rt; a¶rt;um¶rt; n u n¶rt;a ma am mu uu, u¶rt;a ma¶rt;um am am cnma ¶rt;a au.