The fundamental statistical properties of the Vening-Meinesz and stokes integral transformations

Summary The stochastic properties of the vector function, formed by the components and of the deflection of the vertical and by the height of the geoid , are studied by utilizing the mathematical model in[4]. The properties of the error components of the said vector are also studied and a method is described for comparing the results when the vector function was obtained directly by means of astro-geodetic methods and when the vector function was generated by the Vening-Meinesz and Stokes transformation.     

On the gravimetric undulations of the geoid and the deflections of the vertical

Summary The author mentions the aims of the World-wide gravity project he established in the Ohio State University in Columbus, in 1950. He outlines the practical procedure of the gravimetric computations of the undulationsN and the vertical deflection components and and emphasizes that only by the global international cooperation and additional gravity observations at sea carried out during the last decade it has been possible to gather to Columbus the needed gravity material. Since there exist still large gravimetrically unsurveyed areas it is of vital significance to study what gravity anomalies are best to be used for these regions. The given figures concerning the accuracy of theN, and , estimated theoretically and obtained in practice, indicate that in the gravimetrically well surveyed parts of the world like in Europe and the United States we can get gravimetrically on basis of existing gravity material theN-values with accuracy of about 5–10 meters, and and with the accuracy of about 1. The geoid undulationsN are already computed in Columbus for more than 6000 points of the northern hemisphere. The sample maps show the interesting geoid of Europe and vicinity between the latitudes 60° and 30° and longitude 5° W and 30° E, drafted on basis of more than 1000N-values computed at the corners of 1°×1° squares. It is interesting to realize that the geoid undulations in all this area are positive, the extreme values being between 40 and 50 meters. The geophysical significance of the geoid maps of this kind is pointed out.     

A numerical study of the heat transfer through a fluid layer with recirculating flow between concentric and eccentric spheres

A numerical study has been made of the heat transfer through a fluid layer with recirculating flow. The outer fluid surface was assumed to be spherical, while the inner surface consisted of a sphere concentrically or eccentrically located with respect to the outer spherical surface. The recirculating flow was assumed to be driven by a gas flow creating stress on the fluid's outer surface so that creeping (low Reynolds number) flow developed in its interior. The present study solves the Stokes equation of motion and the convective diffusion equation in bispherical coordinates and presents the streamline and isotherm patterns.Nomenclature a _i inner sphere radius - a _d outer sphere radius - A ₁ defined by equation (5) - A ₂ defined by equation (6) - B ₁ defined by equation (7) - B ₂ defined by equation (8) - c dimensional factor for bispherical coordinates - C constant in equation (4) - d narrowest distance between the two eccentric spheres - E ² operator defined by equation (1) in spherical coordinates and by equation (21) in bispherical coordinates - G modified vorticity, defined in equation (22) - G ^* non-dimensional modified vorticity, defined in equation (28) - h metric coefficient of bispherical coordinate system, defined in equation (18) - k _w thermal conductivity of water - K ₁ defined by equation (9) - K ₂ defined by equation (10) - N _Re Reynolds number=2a _dU/gn - N _Pe,h Peclet number=2a _dU/ - n integer counter - q heat flux - r radius - r ^* non-dimensional radius=r/a _d - S surface area - t time - t ^* non-dimensional time=t/a _d ² - T temperature - T _o temperature at inner sphere surface - T _a temperature at outer sphere surface - T ^* non-dimensional temperature;=(T–T _o)/(T_a–T_o) - u velocity - u _r radial velocity in spherical coordinates - u angular velocity in spherical coordinates - u radial velocity in bispherical coordinates - u angular velocity in bispherical coordinates - U free stream velocity - u _r ^* =u _r/U - u ^* =u /U - u ^* =u /U - u ^* =u /U Greek symbols a ₁ small displacement - vorticity, defined in equation (17) - ^* non-dimensional vorticity, defined in equation (27) - radial bispherical coordinates - _o bispherical coordinate of inner sphere - _a bispherical coordinate of outer sphere - angular coordinate in spherical coordinates - thermal diffusivity - _w thermal diffusivity of water - kinematic viscosity - angular bispherical coordinate - spherical coordinate - streamfunction - non-dimensional streamfunction for spherical coordinates, = /(U a _d ² ) - ^* non-dimensional streamfunction for bispherical coordinates, defined in equation (26)     

La risoluzione dell'equazione di Laplace nel campo esterno a uno sferoide

Riassunto L'Autore dimostra che, nel sistema di coordinate polari , , , si possono determinare un numeros di funzioni della sola variabile :Q ₁,Q ₃, ....Q _2s–1 tali che la sommatoria delleQ _2i–1/^2i–1 rappresenti il potenzialeV di un geoide di rotazione. La condizione di armonicità determina ciascunaQ (che si riduce a un polinomio nelle potenze di sen ) a meno di una costante arbitraria; si dispone pertanto dis costanti che servono per soddisfare la natura dellaV sulla superficie del geoide. Come esempio l'Autore ha determinato la gravità sul geoide sferico, confermando i risultati delSomigliana, e su uno sferoide generico dove ha ritrovato la relazione diClairaut.

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Summary The Author proofs that, in the system of polar coordinates , , , it is possible to determine a numbers of functions only of the variable :Q ₁,Q ₃ ....Q _2s–1 in such a way as to make the summatory of theQ _2i–1/^2i–1 represent the potential function of a rotational geoid. The condition of harmonicity determines, saving an arbitrary constant, each of theQ which is reduced to a polynom developed by the sin powers; therefore one disposes of a number of constants to make use for satisfing theV on the geoid. To illustrate his theory the Author determines the gravity on the spherical geoid, thus confirmingSomigliana's formulas and on a spheroidal on which he pointed outClairaut's relations.

     

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     

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     

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 .

     

Die Bestimmung einer beliebig gekrümmten Schichtgrenze aus seismischen Reflexionsmessungen

Zusammenfassung Es wird dargelegt wie man eine beliebig gekrümmte Schichtgrenze aus seismischen Reflexionsmessungen längs einer beliebig gekrümmten Messfläche berechnen kann. Dazu wird vorausgesetzt, dass die Flächez=f(x,y), die die Schichtgrenze darstellt und die Messflächez=(x,y) zusammen eine glatte geschlossene Fläche bilden und, dass das eingeschlossene Material homogen und isotrop ist.

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Summary Under the assumption that the measuring surfacez=(x,y) and the reflection horizon of a structurez=f(x,y) have an arbitrary curvature a simple method is developed for calculating the surfacef from seismic reflection data measured along the surface . In addition the general solution is discussed and some special cases are treated.

     

Study of the velocity conditions in the earth's crust in the regions of the Bohemian massif and the carpathian system along international profiles VI and VII

u¶rt;m n uu ¶rt;u m n u ma n¶rt;aa, nu m¶rt;u u ¶rt;uau. n nuu uu ¶rt;u m n ¶rt;a¶rt; nu NoNo VI, VII. u au u n m a (x, H), ¶rt;auu an¶rt;u ¶rt;u m ¶rt; m¶rt; nu, u mua m nuu (H), aamuu an¶rt;u m a mua ¶rt;¶rt; uu ¶rt;uua u a. u a ¶rt;u m (x, H) amm ¶rt;uuu (u) a m¶rt; nu. m¶rt; u, auuu aau, om aamuam muau an¶rt;u ¶rt;u m, uu , n m u m nu aamuu a nmu, am ma a¶rt;am aa uu ¶rt;u m. aa u a¶rt;u n nu NoNo VI u VII u umuu ¶rt;a. mua m nu (H) num m ¶rt;u amu aua u anam um, ¶rt;a amu ¶rt;aua u ¶rt;- nma (nu No VII). ma ¶rt; ¶rt; aua, maa numa nm m u numa nm ma¶rt;um, a unaa nuu umnmauu a¶rt;uu ¶rt;uau amu aua.     

Deformation of a homogeneous gravitating sphere by internal dislocations

Summary An explicit solution is obtained for the system of equations describing the spheroidal motion in a homogeneous, isotropic, gravitating, elastic medium possessing spherical symmetry. This solution is used to derive the Green's dyad for a homogeneous gravitating sphere. The Green's dyad is then employed to obtain the displacement field induced by tangential and tensile dislocations of arbitrary orientation and depth within the sphere.Notation G Gravitational constant - a Radius of the earth - A _o =4/3 G - Perturbation of the gravitational potential - Circular frequency - V _p ,V _s Compressional and shear wave velocities - k _p =/V _p - k _s =/V _s - k _p [(2.8)] - , [(2.17)] - f _l ⁺ Spherical Bessel function of the first kind - f _l ^– Spherical Hankel function of the second kind - x =r - y =r - x _o =r _o - y _o =r_o - x =r k _s - y =r k _p - x _o =r _o k _s - y _o =r _o k _p - =a - =a - [(5.17)] - _{m, l}      

Rayleigh waves in an inhomogeneous medium

Summary Dispersion in Rayleigh waves is discussed for semi-infinite media with = ₁(1 ± cos s z) and = ₁(1 ± cosh s z), being the rigidity of the medium. A few workers tried with the above Fourier type of model but failed to find the dispersive nature. Because they neglected s due to the complexity of the calculation they arrived at a non dispersive frequency equation. This difficulty is removed in this paper and a dispersive frequency equation is obtained which shows both direct and inverse dispersion. The second model leads to non-convergent solution forz but shows many interesting results which are also discussed.     

Day-time ionospheric disturbances of corpuscular origin in the midlatitudeD-region

¶rt;m uu nau mu m a nu a¶rt;u ¶rt; D-amu u. a¶rt; m nu u u. u¶rt;a a a mu nma u nma mu m (20 ¶rt; 150 ).     

Electromagnetic scattering by a perfectly conducting half-plane in a conductive half-space

FollowingDmitriev (1960) a rigorous theoretical solution for the problem of scattering by a perfectly conducting inclined half-plane buried in a uniform conductive half-space has been obtained for plane wave excitation. The resultant integral equation for the Laplace transform of scattering current in the half-plane is solved numerically by the method of successive approximation. The scattered fields at the surface of the half-space are found by integrating the half-space Green's function over the transform of the scattering current.The effects of depth of burial and inclination, of the half-plane on the scattered fields are studied in detail. An increase in the depth of burial leads to attenuation of the fields. Inclination introduces asymmetry in the field profiles beside affecting its magnitude. Depth of exploration is greater for quadrature component. An interpretation scheme based on a phasor diagram is presented for the VLF-EM method of exploration for rich vein deposits in a conductive terrain.List of symbols x, y, z Space co-ordinates - Half-space conductivity - ₀ Free-space permeability - Excitation frequency (angular) - T Time - h Depth of the half-plane - a Inclination of the half-plane - E _x x-Directed total electric field - E _x ^p x-Directed primary electric field - E _xo ^p x-Directed primary electric field atz=0 directly over the half-plane - H _y y-Component of total magnetic field - H _y ^p y-Component of primary magnetic field - H _y0 ^p y-Component of primary magnetic field atz=0 directly over the half-plane - H _z z-Component of total magnetic field - H _z ^p z-Component of primary magnetic field - J _x Surface density ofx-directed scattering current - G Green's function - k ₀,K Wave numbers - u,u ₀,u ₁,u ₂ Functions - Space co-ordinate - s Variable in transform domain - Variable of integration - Normalized scattering current - Laplace transform of - N Normalized - , ₀, ₁, ₂ Functions - t Variable of integration - Skin depth - H Total magnetic field - H ^p Primary magnetic field - H ₀ ^p Primary magnetic field atz=0 directly over the half-plane - M,Q,R,S,U,V Functions - N ₁,N ₂ Functions     

Accuracy in measurements and the temperature and volume dependence of thermoelastic parameters

To obtain the temperatureT and volumeV (or pressureP) dependence of the Anderson-Grüneisen parameter _T , measurements with high sensitivity are required. We show two examples:P, V, T measurements of NaCl done with the piston cylinder and elasticity measurements of MgO using a resonance method. In both cases, the sensitivity of the measurements leads to results that provide information about _T (,T), where V/V ₀ andV ₀ is the volume at zero pressure. We demonstrate that determination of _T leads to understanding of the volume and temperature dependence ofq=( ln / lnV) _T over a broadV, T range, where is the Grüneisen ratio.     

Connection between the upper ionospheric region of VLF chorus occurrence and the plasmapause position

Summary The occurrence zone of the VLF chorus in the upper ionosphere appears at L-shells lower than plasmapause position L_pp; with increasing geomagnetic activity the spatial dimension of the zone diminishes, its upper boundary being shifted in correspondence with the plasmapause position, the lower remaining practically without change(L=2.0÷2.5). Calculations of propagation paths have shown that the similarity of the VLF chorus spectrum at different upper-ionospheric latitudes as well as the large spatial dimension of the zone of observation can be explained as special features in the propagation of VLF waves from an equatorial source, starting in the vicinity of the plasmapause with different initial normal angles.

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a umauu u ana amu L-, u L_pp (L_pp — nu nana); uu aum amumu nmam a am, nu aua am mmmuu uu nu nana, a u mam namuu u(L=2.0÷2.5). am mamu naam, m n¶rt;u nm a au uma u, ma a nmam ama a¶rt;u, m m mu anmau m amua umua, an amu nana, uu au au.

     

Expansion of a function given over an equipotential surface into spherical harmonics

¶rt;m ¶rt; uuauu uum auu uu, a¶rt;a a nmunmuaa, ¶rt; n uu u. aa, m u u ¶rt;um nmu, uu n¶rt; m nm, u aaaa u aa m nmua.     

On the accuracy of location of local shocks on the territory of Czechoslovakia and adjacent areas

Summary The calculation procedures for determining epicentre parameters of weak near shocks with foci in Poland are discussed and tested for explosions with known epicentres.

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m m¶rt; ¶rt; n¶rt;u num a uu m num nmua n auauu, u mu a mumuu u, n muu ¶rt;au uu mau. au mam nam (a. 4) nu nuuu na 71 u m ¶rt;u n¶rt; ¶rt; a auu ¶rt;a [11].

     

Flow of non-Newtonian fluid through circular tubes

Summary According to Newton's law of viscosity _y = Dv_y/dy. But experiments have shown that _y is indeed proportional to –dv _x/dy for all gases and for homogeneous nonpolymeric liquids. There are however, a few industrially important materials, e.g. plastics, asphalts, crystalline materials that are not described by the equation given by Newton's law of viscosity and they are referred to as non-Newtonian fluids. The steady state rheological behaviour of most fluids can be expressed by the generalised form, _y = –(dv_y/dy) where may be expressed as a function of eitherdv _x/dy or _y (where is independent of the rate of shear, the behaviour is Newtonian with =). Numerous empirical equations or models have been proposed to express the steady-state relation between _y anddv _x/dy. The flow of Newtonian fluids through circular tubes have been discussed before by many. Here we shall discuss the case of two such models of non-Newtonian fluids through circular tubes. The flow of fluids in circular tubes is encountered frequently in Physics, Chemistry, Biology and Engineering.     

On the efficiency with which aerosol particles of radius less than 1 μm are collected by columnar ice crystals

A theoretical model is presented which allows computing the efficiency with which aerosol particles of 0.001 r1 m are collected by columnar ice crystals in air of various relative humidities, temperatures, and pressures. Particle capture due to Brownian diffusion, thermo- and diffusiophoresis is considered. It is shown that phoretic effects importantly determine the particle capture process of 0.01r1 m. The various pressure-temperature levels studied are found to affect the collection efficiency only ifr>0.1 m. Comparison shows that water drops generally are better aerosol particle scavengers than columnar ice crystals.     

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