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).     

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     

Propagation of surface waves on the surface of a non-homogeneous aeolotropic cylindrical shell

Summary The object of the present paper is to investigate the propagation of surface waves on a non-homogeneous aeolotropic cylindrical shell surrounded by vacuum. The elastic constantsc _ij (i, j=1,2...) and density of the material of the shell are assumed to be of the form and respectively, where _ij, ₀ are constants andk ₁,k ₂ are any integers.     

Moto di un vortice sulla Terra e sua influenza sulla polodia

Riassunto Si suppone la Terra avvolta da un velo di un fluido perfetto incomprimibile messo in rotazione da un vortice doppio puntiforme. Si calcola l'energia cinetica totale della Terra e del fluido in funzione degli angoli di Eulero , , , che esprimono il moto della Terra rispetto a una terna inerziale, e degli angoli ₀, ₀ esprimenti il moto del vortice rispetto alla Terra. Si determinano i predetti angoli in funzione del tempo mediante le equazioni di Lagrange; risulta che il moto del vortice è caratterizzato da ₀= const., e che la sua influenza sulla polodia è trascurabile.

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Summary Supposing the Earth sorrounded by a veil of an incompressible perfect fluid rotationally moved by a point shaped double vortex, the Author calculates the total kinetic energy of the system as a function of the Eulerian angles , , which expres the Earth motion referred to an inertial tern, and of the angles ₀, ₀ for the vortex motion referred to the Earth. He determines the above said angles as temporal functions by means of the equations of Lagrange. It results that the vortex motion is determined by ₀= const., and that its influence on the rate of rotation of the Earth is negligeable.

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Comunicazione presentata alla 2^a. Assemblea annuale della «Società Italiana di Geofisica e Meteorologia» (Genova, 23–25 Aprile 1954).     

Strong Potential Magnetic Field Influence on Slightly Differentially Rotating Spherical Shell

An electrically conducting viscous fluid-filled spherical shell is permeated by an axisymmetric strong potential magnetic field with large Elssaser number ² 1. We describe analytically the steady flow driven by a slightly faster rotation of the conducting inner boundary of the shell. The main flow is controlled by Ekman-Hartmann boundary layers with a small thickness /, where ² is the Ekman number. Asymptotics based on small ^–1 1 reveal the nature of a free shear layer O((/)^1/2) and a super-rotation that allows a part of the fluid to rotate faster than the inner sphere. The free shear is following an imposed field line that is tangent to the inner or outer sphere. Meridional flux is concentrated in the shear and boundary layers. Fluid tends to rotate with the inner sphere and to expel azimuthal magnetic field from an -region restricted by the free shear in the spherical shell. For an imposed axial uniform magnetic field, this -region is outside the cylinder tangent to the inner sphere and rotates with the outer sphere. Weak differential rotation O(/) is inside the cylinder, while almost all difference in rotation rates between spheres is accommodated in the thin O((/)^1/2) free shear. For an imposed dipole magnet, the region has a shape of a lobe touching the outer equator. Inside a super-rotation exists; this is the common case for such when the source of the imposed field is inside.     

Magnetic interpretation of dyke anomalies using derivatives

The horizontal and vertical derivative profiles of magnetic anomalies of dykes show some interesting properties. The points of zero derivatives and the points where the derivatives are equal are conjugate point pairs. A method of interpretation of dyke anomalies is suggested, which utilizes the distances between these points.Notation F Magnetic anomaly in total intensity - Z Depth to top of the dyke - 2T Width of the dyke - Geological dip of the dyke - I Effective intensity of magnetisation in the plane of profile - Dip of effective magnetisation vector in the plane of profile - Strike angle of the dyke - i Magnetic dip - Q – - Q _f –+arctan (sin coti) - I _f      

On the free oscillations of a laterally heterogeneous earth model

Summary The equations of motion for the free oscillations of a heterogeneous spherical earth model are derived. It is found that the lateral variations of density and elastic moduli couple the odd(even) harmonics of the spheroidal oscillations with themselves as well as with the even (odd) harmonics of the torsional oscillations.List of symbols r, , Spherical coordinates;r is the radial distance from earth's center, is the co-latitude, and is the east longitude - r Space vector denoting a point with coordinatesr, , and - Gradient operator - ² Laplacian operator - _ij Kronecker's delta function - I Identity matrix - i      

Recherches complémentaires sur le bases du calculateur météorologique «Temp»

Résumé La formule de base, traduisant une propriété analytique d'une classe très générale de fonctions, est un corollaire du théorème fondamental démontré dans un mémoire précédent, d'après lequel, étant donnés une fonction continue,p(, ,t) des points (, ) d'une surface régulière fermée et du temps et le champ d'un vecteur vitesse de transfert ou d'advection tangent à et ayant des lignes de flux fermées et régulières, il existe un opérateur spatial, linéaire, non singulierA tel que la fonctionA(p+Const.) soit purement advective par rapport a (sans creusement ni comblement). Ce théorème peut être exprimé par l'équation , où est un opérateur spatial, linéaire et non singulier, fonction deA.La détermination de peut être faite, soit en comparant deux formes différentes de la solution générale de l'équation en , soit en utilisant un raisonnement a priori très simple. On arrive ainsi au résultat pour un certain scalaireu(, ).Dans le cas oùp(, ,t) est la perturbation de la pression sur la surface du géoïde l'équation résulte aussi, comme nous l'avons montré dans le mémoire précédent, de notre théorie hydrodynamique des perturbations. On montre ici que la même équation peut encore être déduite de l'équation de continuité associée à la condition d'équilibre quasi statique selon la verticale.Comme applications de la formule de base (solution générale de l'équation enM), on étudie les problèmes suivants: 1^o creusement et comblement en général; 2^o creusement et comblement des centres et des cols; 3^o mouvement des centres et des cols; 4^o instabilité d'un champ moyen; 5^o propriétés spatiales des champsp(, ,t) et des vecteurs d'advection analytiques.Après une discussion des erreurs de la prévision d'un champp(, ,t) par la formule de base, du fait des erreurs des observations et du fonctionnement du calculateur, on examine quelques particularités du transfert ou advection d'un champf ₀(, ) par le vecteur . Enfin, le dernier chapitre du mémoire donne des éclaircissements complémentaires sur la structure du calculateur électronique «Temp» (qui effectue automatiquement les opérations mathématiques de la formule de base) et expose l'état actuel de sa construction.

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Summary The basic formula, expressing an analytical property of a very general class of functions, is a corollary of the fundamental theorem, proved in a previous paper, according to which, given a functionp(, ,t) of the points (, ) of a closed regular surface and of the time, and a transfer or advection velocity vector tangent to and having regular closed streamlines, there is a spatial, linear, non singular operatorA such thatA(p+const.) is a purely advective function in respect to (no deepening). This theorem can be expressed by the equation where is a spatial, linear, non singular operator depending onA.The determination of can be attained, either by the comparison of two different forms of the general solution of the -equation, or by a simple a priori reasonning. The conclusion is thus reached that for a certain scalaru(, ).Whenp(, ,t) is the pressure perturbation at sea level, it was shown, in the preceding paper, that the equation can also be derived from our hydrodynamical perturbation theory. We now show that for this particular case, the same equation is also a consequence of the equation of continuity together with the condition of quasi statical vertical equilibrium.The following problems are then analysed by means of the basic formula: 1^o deepening and filling in general; 2^o deepening and filling of the centres and cols; 3^o motion of the centres and cols; 4^o instability of a mean field; 5^o spatial properties of the analytical fields and advection vectors .The errors in the forecast of a field,p(, ,t) by means of the basic formula, due to the observational and computational errors, are discussed, and some peculiarities of the transfer or advection of a fieldf ₀(, ) by are examined. Finally, complementary points are disclosed on the structure of the electronic computer «Temp» which performs automatically the mathematical operations of the basic formula, and a brief report is given of the present state of its construction.

     

Long-period fluctuations observed in the ionospheric absorption variations

Summary By subsequent application of power spectrum analysis, autocorrelation analysis and fast Fourier transform (FFT) of the day- and night-time absorption values of five LF radio-paths (164 kHz, 155 kHz, 185 kHz, 218 kHz and 272 kHz) in Europe during the interval 1 June 1979–30 June 1980, fluctuations with the following basic (fundamental), commonly recurrent periods were found: 3.5–5 days and 10.5–12 days. They exist in all investigated time series, while 6- and 9-day fluctuations are observed on the northern radio-paths (185 kHz and 272 kHz). Shorterperiod oscillations are most active during autumn and especially during winter, while the longerperiod oscillation (10.5–12-day) has significant amplitudes also in summer.

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n n¶rt;am nuu aaua nma mu, amu aaua u m maauu () ¶rt; u ¶rt;a n nu u nmu a¶rt;u ma (164 , 155 , 185 , 218 u 272 ) a¶rt; n nu¶rt; 1 u 1979–30 u 1980. u a¶rt; ¶rt;u (¶rt;ama) nmu nu¶rt;: 3.5–5 ¶rt; u 10.5–12 ¶rt. u nummm u¶rt;a ¶rt;a. 6- u 9-m au a¶rt;am m a¶rt;umaa (185 u 272 ). ama au a amu u u u. ¶rt;nu¶rt; (10.5–12 ¶rt;) um aum anum¶rt; u m.

     

The connection between characteristic values of the 11-year cycles of solar and of geomagnetic activity

Summary Using the annual values of the indices of solar and geomagnetic activity for the period 1868–1976, the basic values characterizing the cycle as a whole were determined for the 11-year cycles nos 11–20, (Tab. 1). High values of the coefficients of correlation were found for some pairs of characteristic values of the same and different kinds, given in Tab. 2, which can be utilized for long-term predictions of geomagnetic activity.

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a auu¶rt;u au u¶rt; uaum amumu muu 1868–1976 n¶rt; ¶rt; 11-mu u 11–20 naam aamuu u a (a. 1). u au uuma uu ¶rt; m na naam ¶rt;ua u ¶rt;ua ¶rt;a (a. 2), m m amu nuu nu ¶rt; nuauuaum amumu.

     

Determination of the geopotential scale factor from satellite altimetry

Summary The geopotential scale factor R ₀ =GM/W ₀ has been determined on the basis of satellite altimetry as R _{0=(6 363 672·5±0·3) m} and/or the geopotential value on the geoid W ₀ =(62 636 256·5±3) m ² s ^–2 . It has been stated that R ₀ and/or W ₀ is independent of the tidal distortion of surface W=W ₀ due to the zero frequency tide.

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¶rt;a nmu amumuu u ama amnmuaa R ₀ =GM/W ₀ =(6 363 672,5±0,3) m u/uu aunmuaa a nmuu¶rt;a W ₀ =(62 636 256,5±3) m² s^–2. m, m R ₀ u/uu W ₀ auum m nm amu a a nuu ¶rt;au nmu W=W ₀ .

     

Zur Interpretation von seismischen Refraktionsmessungen

Zusammenfassung Unter der Voraussetzung, dass die Frontgeschwindigkeitc eine stetige und monoton wachsende Funktion von der Tiefez ist, wird dargelegt, wie man aus einer gemessenen Laufzeitkurve () die zuc inverse Funktionz=z (c) auf einfache Weise berechnen kann. Weiter wird die Eindringtiefez _m in Funktion von ermittelt und abschliessend ein Beispiel gegeben.

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Summary Based on a recorded travel-time curve (), a simple direct method is developed for calculating the functionz=z (c), under the asumption that the wave velocityc is a regularly monotone increasing function of the depthz. Finally a rumerical example is given.

     

Typical properties of dynamic systems and the instability of hydrodynamic flows

Summary Based on Peixot's theorem of topological dynamics, the unstable behaviour of hydrodynamic flows on a two-dimensional annuloid T ² is analysed. The generating property is the curvature of the group of S ₀ Diff T ² diffeomorphisms of the (Riemannian) flow region T ² . This group is the configuration space of the ideal fluid flowing on T ² .

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a m ma u mn u ¶rt;uauu ¶rt;am mu n¶rt;uu¶rt;¶rt;uauu nm a ¶rt; m T ² u¶rt; mu. ¶rt;au m m a m uuann S ₀ Diff T ² ¶rt;uu amu mu ( uaa) T ² . manna m uau nmam ¶rt; u¶rt;a u¶rt;mu a T ² .

     

Source parameters of some large earthquakes in the eastern Mediterranean region based on an itertive maximum entropy technique

TeleseismicP-waves of some large earthquakes that occurred in the eastern Mediterranean region have been analysed by using an iterative maximum entropy technique in order to obtain the independent spectral parameters, the long-period spectral level ₀ and the corner frequencyf ₀ of the far-field displacement spectra.Based on these parameters, the seismic source parameters seismic momentM ₀, source dimensionr, fault lengthl, average displacement u, shear stress drop , radiated energyE _s and apparent stressn are calculated for the considered earthquakes by using Brune's and Madariaga's models.The striking feature of the source parameters obtained in this study is the low stress drop value which varies between 5 and 15 bars. If Madariaga's model had been used, higher stress drop values would have been obtained.The low stress drop earthquakes in the eastern Mediterranean region might be interpreted either by the possible presence of low strength material near the source or by the partial stress drop model.     

Single- and inter-station transfer functions for non-synchronous geomagnetic arrays — I. Theory of methods used in data analysis

Summary The theory of the multivariate coherence analysis(spectral domain approach) is developed for calculating single- and inter-station transfer functions and corresponding vector induction characteristics from time variations of the geomagnetic field components. An alternative approach of calculating similar induction characteristics using a time domain algorithm is shown.

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aam mu m aaua(nma n¶rt;¶rt;) nuuu ama ¶rt;-u -mau n¶rt;am u u mmmu m aamumu u¶rt;uu n anua auau mau aum n u. u¶rt;um ma m¶rt; u aau aamumu u¶rt;uu nm ama amu un mua um. nuam aum na auu am u aamum ¶rt; n ama naam.

     

A model for the simulation of turbulent and eddy diffusion processes at heights of 0–65 km

A generalized turbulent diffusion model has been developed which evaluates the time rate of growth of a simulated cloud of particles released into a turbulent (i.e. diffusive) atmosphere. The general model, in the form of second-order differential equations, computes the three-dimensional size of the cloud as a function of time. Parameters which influence the cloud growth, and which are accounted for in the model equations, are: (1) length scales and velocity magnitudes of the diffusive field, (2) rate of viscous dissipation , (3) vertical stability as characterized by the relative adiabatic lapse rate (1/T)(g/C _p +T/z), and (4) vertical shear in the mean horizontal winds , and , for a given height and of spatial extent equal to that of the diffusing cloud. Sample results for near ground level and for upper stratospheric heights are given. For the atmospheric boundary layer case, the diffusive field is microscale turbulence. In the upper stratospheric case it is considered to be a field of highly interactive and dispersive gravity waves.     

Magnetoconvection in Dependence on Roberts Number

Marginal instabilities in a horizontal unstably stratified rapidly rotating fluid layer permeated by an azimuthal magnetic field growing linearly with distance from the vertical rotation axis, are investigated in dependence on electromagnetic boundary conditions and Roberts number . Electrical conductivities of perfectly thermally conducting boundaries are either infinite or finite Earthlike ones. Following Soward's (1979) results the case of gives the instabilities of MAC-waves with much greater frequencies than for the case of q 1. Critical frequencies of MAC waves are most sensitive on Roberts number for q 1 and lose this sensitivity for q2. On the other hand the critical Rayleigh numbers are slightly dependent on q 1. Many qualitative differences in the dependences on q do exist for infinitely and finitely conducting boundaries. The former do not allow fast eastward modes for q 1 and the latter, e.g. allow eastward MC-waves with azimuthal wave number m = 1. The MC-waves are almost independent on Roberts number q.     

On the possibility of utilizing grid reports in a local model of short-range forecast of meteorological elements

Summary In the experiment described, we test the possibility of utilizing forecasts of the pressure field, contained in GRID reports, to solve the problem of horizontal boundary conditions of a local model of short-range forecast of meteorological elements. We prove that the assumption of a linear tendency of the prognostic variables in the boundary region yields good results, using the Perkey-Kreitzberg method[1], even if applied to period T=24 hrs. In the Perkey-Kreitzberg method the effect of the horizontal diffusion in the boundary region is suppressed[2]. However, considering the diffusive term apart from the procedure of the method mentioned proves detrimental to the forecast.

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nua num n m unau n n ¶rt;au, ¶rt;au ¶rt; ¶rt; u numaau u a ¶rt;u am na muu m. aam, m n¶rt;nu u m¶rt;uu nmuu n au amu nu unauu m¶rt;a u- [1] ¶rt;am u mam ¶rt;a nu nuuu nu¶rt; T=24 a. m¶rt; u- nu¶rt;um n¶rt;au uuuma ¶rt;uuu au amu[2]. ama ¶rt;u a n¶rt; nu¶rt; m¶rt;a nu¶rt;um, ¶rt;a, ¶rt;u n.

     

Propagation velocity and radiation properties of induced tensile cracks

nua ¶rt;u a¶rt;am m aa ¶rt;uaaa . ¶rt;u aam ¶rt; ¶rt; uu umuo au uu m a u a u am mu ma. mu anmau mu ma n¶rt; auumu m a aa u anau nu au. num ma uu nu un u n mu ma n¶rt;mauu n¶rt;um uu m umua. mam ¶rt;umum mu u aua n au muna ma a umua uu un. ¶rt; aauua am na mu ma, m uam ua u.     

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.