Propagation of love waves in a non-homogeneous stratum of finite depth sandwitched between two semi-infinite isotropic media

Summary The aim of this paper is to study a problem in which the intermediate layer is non-homogeneous, the rigidity varying exponentially with depth i.e. ₂=₂ v ₀ ² e ^2pz , the density being constant, velocity varies also exponentially with depth according to the law =v ₀ e ^pz . The variability ofKH with the change of phase velocity is shown graphically.     

Sull'approssimazione numerica di una variabile casuale normale,con applicaztone agli errori d'osservazione

Riassunto Data una variabile casuale X che segue la legge normale di probabilitl con valor medio a ed error medio y 1'A. considera un'altra variabile casuale che prende il valore intero r quando r–1/2

Summary Given a random variable X following the normal probability law, with expectation a and standard error p, the author considers another random variable , that takes the entire value r when r–1/2     

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.     

Relations entre le creusement et le comblement des perturbations et leur déplacement

Résumé On commence par définir le creusement et le comblement d'une fonctionp(, t) du tempst et des points (, ) d'une surface régulière fermée en se donnant, sur cette surface, un vecteur vitesse d'advection ou de transfert tangent à . Le creusement (ou le comblement) est la variation dep sur les particules fictives se déplaçant constamment et partout à la vitesse , A chaque vecteur et pour un mêmep(, ,t) correspond naturellement une fonction creusementC (, ,t) admissible a priori; mais une condition analytique très générale (l'intégrale du creusement sur toute la surface fermée du champ est nulle à chaque instant), à laquelle satisfont les fonctions de perturbation sur les surfaces géopotentielles, permet de restreindre beaucoup la généralité des vecteurs d'advection admissibles a priori et conduit à des vecteurs de la forme: , oùT est un scalaire régulier, () une fonction régulière de la latitude , le vecteur unitaire des verticales ascendantes etR/2 une constante. Ces vecteurs sont donc une généralisation naturelle des vitesses géostrophiques attachées à tout scalaire régulier. Dans le cas oùp(, ,t) est la perturbation de la pression sur la surface du géoïde, le vecteur d'advection par rapport auquel on doit définir le creusement est précisément une vitesse géostrophique: on a alors ()=sin etT un certain champ bien défini de température moyenne.On déduit ensuite une formule générale de géométrie et de cinématique différentielles reliant la vitesse de déplacement d'un centre ou d'un col d'un champp(, ,t) à son champ de creusementC (, ,t) et au vecteur d'advection correspondant. Cette formule peut être transformée et prend la forme d'une relation générale entre le creusement (ou le comblement) d'un centre ou d'un col et la vitesse de son déplacement, sans que le vecteur d'advection intervienne explicitement. On analyse alors les conséquences de ces formules dans les cas suivants: 1^o) perturbations circulaires dans le voisinage du centre; 2^o) perturbations ayant, dans le voisinage du centre, un axe de symétrie normal ou tangent à la vitesse du centre; 3^o) évolution normale des cyclones tropicaux.Finalement, on examine les relations qui existent entre le creusement ou le comblement d'un champ, le vecteur d'advection et la configuration des iso-lignes du champ dans le voisinage d'un centre.Ces considérations permettent d'expliquer plusieurs propriétés bien connues du comportement des perturbations dans différentes régions.

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Summary The deepening and filling (development) of a functionp(, ,t) of the timet and the points (, ) of a regular closed surface is first of all defined, in respect to a given advection or transfer velocity field tangent to , as the variation ofp on any fictitious particle moving constantly and everywhere with the velocity . For a givenp(, ,t) and to any there corresponds a well defined development fieldC (, ,t). All theseC fields are a priori admissible, but a very general analytical condition of the perturbation fields in synoptic meteorology (the integral of the development fieldC (, ,t) on any geopotential surface vanishes at any moment), leads to an important restriction to advection vectors of the form: , whereT is any regular scalar, () any regular function of latitude, the unit vector of the ascending verticals andR/2 a constant. These vectors are a natural generalisation of the geostrophic velocities attached to any regular scalar. Whenp(, ,t) is the pressure perturbation at sea level, its development must be defined in respect to a geostrophic advection vector belonging to the above defined class of vectors with ()=sin andT a well defined mean temperature field.A general formula of the differential geometry and kinematics ofp(, ,t) is then derived, giving the velocity of any centre and col of ap(, ,t) as a function of the advection vector and the corresponding development fieldC (, ,t). This formula can be transformed and takes the form of a general relation between the deepening (and filling) of a centre (or a col) of ap(, ,t) and its displament velocity, the advection vector appearing no more explicitly. A detailed analysis of the consequences of these formulae is then given for the following cases: 1^o) circular perturbations in the vicinity of a centre; 2^o) perturbations having, in the vicinity of a centre, an axis of symmetry normal or tangent to the velocity of the centre; 3^o) normal evolution of the tropical cyclones.Finally, the relations between the developmentC (, ,t) of a fieldp(, ,t), the advection velocity vector and the configuration of the iso-lines in the vicinity of a centre are analysed.These theoretical results give a rational explanation of several well known properties of the behaviour of the perturbations in different geographical regions.

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Communication à la 2ème Assemblée de la «Società Italiana di Geofisica e Meteorologia» (Gênes, 23–25 Avril 1954).     

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).     

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).     

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.

     

Ueber Herde elastischer Wellen in isotropen,homogenen Medien

Zusammenfassung 1) Es werden Multipollösungen der skalaren Wellengleichung ² f/t ² – c² div gradf=0 betrachtet. Einerseits kann man solche Lösungen direkt durch Kugelfunktionenn-ter Ordnung ausdrücken, anderseits aus der Einpollösungf=1/p F(t–p/c) durch Differentiation nachn Richtungen erhalten. Es wird der Zusammenhang zwischen den Ergebnissen der beiden Verfahren gezeigt. — 2) Für die Energiedichte und den Energiefluss durch Kugelflächen bei kleinen elastischen Verschiebungen werden Ausdrücke in Kugelkoordinaten angegeben. — 3) Für die Wellengleichung grad div –b ² rot rot werden rotationsfreie Multipollösungen angegeben und Ausdrücke für Energiedichte und Energiefluss hergeleitet. — 4) Das gleiche wird für divergenzfreie Multipollösungen durchgeführt. — 5) Es werden Multipole betrachtet, die weder rotationsfrei noch divergenzfrei sind. Als Spezialfälle werden Multipole mit zeitlich begrenzter und solche mit periodischer Erregung gezeigt, ferner Lösungen der Wellengleichung, die sowohl rotationsfrei wie divergenzfrei sind. — 6) Es wird gezeigt, wie man die elastischen Wellen, die im Sinne vonStokes von einem Herdgebiet endlicher Ausdehnung ausgehen, näherungsweise durch elastische Multipole darstellen kann. — 7) Es wird angedeutet, wie man durch Messung von Komponenten von oder u.s.w. in Punkten im Innern des Mediums die Erregung und Energie von elastischen Multipolen bestimmen kann. Ferner wird auf den Fall hingewiesen, wo ein rotationsfreier Einpol sich im Innern eines Halbraumes befindet und die Messungen an seiner Oberfläche ausgeführt werden.

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Summary (On foci of elastic waves in isotropic homogeneous media) — 1) Multiplets as solutions of the scalar wave equation ² f/t ² – c² div gradf=0 are considered. Such solutions can be obtained either directly by aid of spherical harmonics of ordern, or by differentiating the single polef=1/p F(t–p/c) with respect ton directions. The relations between the results of those two procedures are shown. — 2) In the case of small elastic displacements , the density of energy and the flow of energy through spherical surfaces are expressed by spherical coordinates. — 3) Multiplets which satisfy the equation of motion =a ² grad div b ² curl curl and the equation curl = 0 are given, and expressions for the density and flow of energy are found. — 4) The same is done with multiplets satisfying the equation of motion and the equation div = 0. — 5) General multiplets which satisfy the equation of motion are treated. As special cases, multiplets with excitation of finite length and multiplets with periodic excitation are considered, furthermore solutions of the equation of motion and of the equations curl = 0 and div = 0 are given. — 6) It is shown how elastic waves whose origin is a region of finite extension in the sense given byStokes, can be approximated by elastic multiplets. — 7) Some indications are given on the problem of how to find the functions of excitation and the energy of an elastic multiplet by measuring components of or etc., at points in the interior of the medium. The same problem is considered in the case of the single elastic pole. = grad 1/p F (t–p/a), if the measurements are made at the surface of an elastic half space.

     

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.     

Estimate of earthquake hazard in the Vrancea (Romania) region

A maximum likelihood method is used to estimate the earthquake hazard parameters maximum magnitudeM _max, annual activity rate , and theb value of the Gutenberg-Richter equation in the Vrancea (Romania) region. The applied procedure permits the use of mixed catalogs with incomplete historical as well as complete instrumental parts, the consideration of variable detection thresholds, and the incorporation of earthquake magnitude uncertainty.Our imput data, comprises 105 historical earthquakes which occurred between 984 and 1934, and a complete data file containing 1067 earthquakes which occurred during the period 1935–30 August, 1986. The complete part was divided into four subcatalogs according to different thresholds of completeness. Only subcrustal events were considered, and dependent events were removed.The obtained value (=0.65) is at the lower range of the previously reported results, but it appears concurrent with conceptual and observational facts. The same concerns inferred value of _max = 7.8 and activity rate _4.0 = 5.34.     

An algorithm for the optimum distribution of a regional seismic network — II. An analysis of the accuracy of location of local earthquakes depending on the number of seismic stations

Summary Seven optimal networks consisting of 4 to 10 stations are compared for a given region, where velocity-depth profiles and the distribution of seismic intensity are known. Assuming that the standard error of arrival time is _t =0.05 s and the standard errors of the parameters of velocity-depth profiles are equal to 5% of their values, the average standard errors of the origin time and focus coordinates are estimated. The application of optimum methods to the planning of seismic networks in the Lublin Coal Basin is presented, and maps of standard errors of origin time , depth and epicenter ( _xy ) for the case of an optimum network of 6 seismic stations are given.     

“Pandora box” — Matrix in the calculus of observations

Summary If the condition R(A)=k(n), whereA is the design matrix of the type n × k and k the number of parameters to be determined, is not satisfied, or if the covariance matrixH is singular, it is possible to determine the adjusted value of the unbiased estimable function of the parameters f(), its dispersion D( (x)) and ² as the unbiased estimate of the value of ² by means of an arbitrary g-inversion of the matrix . The matrix , because of its remarkable properties, is called the Pandora Box matrix. The paper gives the proofs of these properties and the manner in which they can be employed in the calculus of observations.     

The derivatives of the lunar disturbing potential harmonics with respect to euler's angles

Summary The derivatives of the harmonicsP _n ^(k) (sin _O)cos kT_O andP _n ^(k) (sin _O)sin kT_O, occurring in the development of the lunar disturbing potential, are derived upto n=4 and for k== 0, 1, ..., n. The equatorial co-ordinates _OT_O are referred to the Moon's mass centre; the procedure for the solar disturbing potential is formally identical.     

Some comparisons between mining-induced and laboratory earthquakes

Although laboratory stick-slip friction experiments have long been regarded as analogs to natural crustal earthquakes, the potential use of laboratory results for understanding the earthquake source mechanism has not been fully exploited because of essential difficulties in relating seismographic data to measurements made in the controlled laboratory environment. Mining-induced earthquakes, however, provide a means of calibrating the seismic data in terms of laboratory results because, in contrast to natural earthquakes, the causative forces as well as the hypocentral conditions are known. A comparison of stick-slip friction events in a large granite sample with mining-induced earthquakes in South Africa and Canada indicates both similarities and differences between the two phenomena. The physics of unstable fault slip appears to be largely the same for both types of events. For example, both laboratory and mining-induced earthquakes have very low seismic efficiencies where _a is the apparent stress and is the average stress acting on the fault plane to cause slip; nearly all of the energy released by faulting is consumed in overcoming friction. In more detail, the mining-induced earthquakes differ from the laboratory events in the behavior of as a function of seismic momentM ₀. Whereas for the laboratory events 0.06 independent ofM ₀, depends quite strongly onM ₀ for each set of induced earthquakes, with 0.06 serving, apparently, as an upper bound. It seems most likely that this observed scaling difference is due to variations in slip distribution over the fault plane. In the laboratory, a stick-slip event entails homogeneous slip over a fault of fixed area. For each set of induced earthquakes, the fault area appears to be approximately fixed but the slip is inhomogeneous due presumably to barriers (zones of no slip) distributed over the fault plane; at constant , larger events correspond to larger _a as a consequence of fewer barriers to slip. If the inequality _a / 0.06 has general validity, then measurements of _a =µE _a /M ₀, where is the modulus of rigidity andE _a is the seismically-radiated energy, can be used to infer the absolute level of deviatoric stress at the hypocenter.     

Planar charged-particle trajectories in multipole magnetic fields

This paper provides a complete generalization of the classic result that the radius of curvature () of a charged-particle trajectory confined to the equatorial plane of a magnetic dipole is directly proportional to the cube of the particles equatorial distance () from the dipole (i.e. ³). Comparable results are derived for the radii of curvature of all possible planar chargedparticle trajectories in an individual static magnetic multipole of arbitrary order m and degree n. Such trajectories arise wherever there exists a plane (or planes) such that the multipole magnetic field is locally perpendicular to this plane (or planes), everywhere apart from possibly at a set of magnetic neutral lines. Therefore planar trajectories exist in the equatorial plane of an axisymmetric (m = 0), or zonal, magnetic multipole, provided n is odd: the radius of curvature varies directly as ⁿ⁼². This result reduces to the classic one in the case of a zonal magnetic dipole (n = 1). Planar trajectories exist in 2m meridional planes in the case of the general tesseral (0 < m < n) magnetic multipole. These meridional planes are defined by the 2m roots of the equation cos[m()–_n^m)] = 0, where _n^m = (1/m) arctan (h_n^m/g_n^m); g_n^m and h_n^m denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (n – m) is odd. The polar axis ( = O,) of a tesseral magnetic multipole is a magnetic neutral line if m > I. A further 2_m(n – m) neutral lines exist at the intersections of the 2_m meridional planes with the (n – m) cones defined by the (n – m) roots of the equation P_n^m(cos ) = 0 in the range 0 < 9 < , where P_n^m(cos ) denotes the associated Legendre function. If (n – m) is odd, one of these cones coincides with the equator and the magnetic field is then perpendicular to the equator everywhere apart from the 2m equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to ⁿ⁼² and inversely proportional to cos[m(–)]. Since this last expression vanishes at the 2m equatorial neutral ines, the radius of curvature becomes infinitely large as the particle approaches any one of these neutral lines. The radius of curvature of a meridional trajectory is directly proportional to rⁿ⁺², where r denotes radial distance from the multiple, and inversely proportional to P_n^m(cos )/sin . Hence the radius of curvature becomes infinitely large if the particle approaches the polar magnetic neutral ine (m > 1) or any one of the 2m(n – m) neutral ines located at the intersections of the 2m meridional planes with the (n – m) cones. Illustrative particle trajectories, derived by stepwise numerical integration of the exact equations of particle motion, are pressented for low-degree (n 3) magnetic multipoles. These computed particle trajectories clearly demonstrate the non-adiabatic scattering of charged particles at magnetic neutral lines. Brief comments are made on the different regions of phase space defined by regular and irregular trajectories.Also Visiting Reader in Physics, University of Sussex, Falmer, Brighton, BN1 9QH, UK     

Effect of cofactors of adjusted and unadjusted quantities on their functions

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Linear stability of a non-constantly stratified horizontal layer penetrated by an azimuthal magnetic field

Summary The convection in a rapidly rotating, electrically conducting, horizontal fluid layer, non-constantly stratified and penetrated by an inhomogeneous magnetic field, is studied. The convection is investigated for various ratios of the thickness of the stable and unstable stratified part of the layer. The thermal model of the layer, as well as the analysis of the results have been treated with regard to the physical conditions in the liquid core of the Earth.

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am u¶rt;m u m aa mn¶rt; u¶rt;uma nm mamuuau, nua ¶rt;¶rt; aum n. u u¶rt;m ¶rt; a mu m u mu u mu mamuuuao amu . ua ¶rt; , a u aau mam, n¶rt;a anm uuu u u¶rt; ¶rt; u.

     

The dynamics of magma mixing in a rising magma batch

The conditions under which two magmas can become mixed within a rising magma batch are investigated by scaling analyses and fluid-dynamical experiments. The results of scaling analyses show that the fluid behaviours in a squeezed conduit are determined mainly by the dimensionless number where ₁ is the viscosity of the fluid, U is the velocity, g is the acceleration due to gravity, is the density difference between the two fluids, and R is the radius of the tube. The parameter I represents a balance between the viscous effects in the uppermost magma which prevent it from being moved off the conduit walls, and the buoyancy forces which tend to keep the interface horizontal. The experiments are carried out using fluid pairs of various density and viscosity contrasts in a squeezed vinyl tube. They show that overturning of the initial density stratification and mixing occur when I>order 10^-1; the two fluids remain stratified when I 10^-3. Transitional states are observed when 10^-3<I<10^-1. These results are nearly independent of Reynolds number and viscosity ratio in the range of and Re ₁<300. Applying these results to magmas shows that silicic to intermediate magmas overlying mafic magma will be prone to mixing in a rising magma batch. This mechanism can explain some occurrences of small-volume mixed lava flows.     

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      

Relation between the field of surface heat flow and the distribution ofP n -wave velocities for continents

Summary The dependence between P_n-wave velocities and the surface heat flow, temperature at the core-mantl boundary and thickness of the Earth's crust for continents (Europe, Asia, North America and Australia) was investigated statistically in connection with the problem of lateral inhomogeneities in the upper mantle. The relations obtained were compared with those determined under laboratory conditions. The conclusion is that temperature and pressure effects may provide additional explanations of the regional variations of P_n-wave velocities observed in most continents.

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auum ¶rt;auu mu n¶rt; a nmu uua(P_n ), nm mn nm, mnam a u m mum a u¶rt;aa u n uuuma ¶rt;¶rt;m mu P_n. nua ¶rt;a mama aam u¶rt;au nu m n¶rt; amuu u u ¶rt;au u mnam a¶rt;um mmmuu mamau n¶rt;aa am. am ¶rt;, m ua uu m P_n- ¶rt; amu muma n¶rt;m auu m¶rt;uauu u a nmu muua.