Transmission of Rayleigh waves over the surface of a heterogeneous medium

Summary The frequency equation of Rayleigh waves propagating over the free surface of an isotropic, perfectly elastic, heterogeneous semi-infinite medium with material properties varying as = ₀ e ^az , = ₀ e ^az , = ₀ e ^az (a>0) has been obtained. Solution of the frequency equation in closed form is obtained in two cases (i) =0, (ii) =, and the Rayleigh wave dispersion curves for phase and group velocities drawn. In both the cases the medium yields single Rayleigh modes which cannot propagate below certain cut-off frequencies. It is found that in case (i), <c<c ₀ and 0.87500 <c _g <c ₀, and in case (ii), 1.03082 <c<c ₁ and 0.90850 <c _g <c ₁, wherec andc _g denote phase nad group velocities respectively, is the constant shear wave velocity of the mediumc ₀ andc ₁ are the corresponding Rayleigh wave velocities of the homogeneous medium of the same Poisson's ratio. The motion of the surface particles is found to be retrograde elliptical as in the homogeneous case, but the ratic of the major and minor axes now becomes frequency dependent and is plotted against frequency. In both the cases (i) and (ii), the ratio starts at a lower value at the cut-off frequency and approaches the corresponding value of the homogeneous medium at high frequencies.     

Observed Response of the Ionospheric F2–Layer and Lower Thermosphere to Geomagnetic Storms during High Solar Activity

The response of the critical frequency of the ionosphere F2–layer, described by its main Fourier components (daily constant, diurnal and semidiurnal waves) and the lower thermosphere dynamics to the geomagnetic storms in July 1991 and February 1992 is studied. The daily constant displays a negative response, however, the magnitude of reaction depends on the season and latitude. The amplitudes of diurnal and semidiurnal waves increase during a geomagnetic storm, as this enhancement is very strong at high latitudes in winter. The prevailing neutral wind, especially the zonal wind, shows an inclination to decrease during the geomagnetic storm (the effect is more distinct in summer). The amplitudes of diurnal and semidiurnal tides also demonstrate a tendency toward reduction during high geomagnetic activity.     

The secular variation of the geomagnetic field in Egypt in the last 5000 years

The palaeo-intensities (F _a) of the geomagnetic field in Egypt at some ages are determined by archaeomagnetic measurements and found to be:F _a=36.2 T at 3100 B.C., F_a=46.8 T at 3000 B.C.,F _a=36.5 T at 2780 B.C., 49.0 T at 2500 B.C., 36.4 T at 2200 B.C., 57.5 T at 1990 B.C., 62.1 T atca 1400 B.C., 61.5 T at 1400 B.C., 69.9 T at 600 B.C., 59.3 T at 550 B.C., 79.9 T at 460 B.C., 73.7 T at 450 B.C., 69.7 T at 320 B.C., 56.2 T at A.D. 50, 64.9 T, at A.D. 400, 54.4 T at A.D. 300, 57.5 T at A.D. 700 and 43.0 T at A.D. 1975.The palaeo-inclinations (I _a) at some ages are found to be:I _a=24.2° at 420 B.C., 44° at A.D. 50, 60.7° at A.D. 703 and 42° at A.D. 1795.The measured values ofF _a are affected by the anisotropy of magnetic susceptibility of the samples by 13% to 20% of the expected correct value. The suitable correction of this effect is by multiplyingF by 1/((1+0.2(/90)) andF by 1/((1–0.13 (/90)), whereF andF are the resultant values ofF _a if the laboratory field is perpendicular or parallel to the wall of the sample during the Thelliers' experiments, respectively, and is the angle between the direction of natural remnant magnetization of the sample and the direction of the laboratory field.The results of this paper, together with the previous results for Egypt and the neighbourhoods, lead to the production of the secular variation curve of the geomagnetic field in Egypt for the last 5000 years. The intensity of the field shows a periodicity of about 400 years with multiples.     

On the further generalization of trochoidal frontal waves

Summary A non-linear model of trochoidal waves is presented which represents a geometrical and kinematical generalization of Gerstner's waves and of the results of[2–4].

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¶rt;aam ¶rt; mu¶rt;a , ma m u ma u am[2–4] mu umuu u uamu mm ¶rt;u.

     

Hm-wave filtration in a horizontally stratified Earth's magnetosphere

Summary Drawing on papers [4, 5] a semi-quantitative model of filtration and resonance of a linearly polarized HM-wave has been constructed in a horizontally stratified Earth atmosphere. The wave is filtered in a system of 3 homogeneous layers, limited by two halfspaces. Attenuation of the wave is only assumed in the lower layer, i.e. in the ionosphere. A relation has been derived for the amplitude coupling factor H_T/H_F expressing the absolute ratio of the amplitude of the wave after transmission through the system of layers and the amplitude of the wave, incident at the system of layers. The mechanism of geomagnetic pulsations is illustrated on the principle of the resonance of a HM-wave in the Earth's magnetosphere. The magnetosphere has been replaced by the said system of layers, and the dependence of H_T/H_F=f() on the parameters of the system of layers has been studied using a computer.     

The role of wavegroup velocity in the blocking problem

Considering the blocking problem as a baroclinic instability problem in a dispersive wave system with diabatic heating effects, it is of great interest to investigate the role of wavegroup velocityv _gr in blocking processes, becausev _gr controls the energy transfer in the wave field. Using a Newtonian Cooling —type of forcing with a phase differencek to the main field and taking the linearized version of a two-level model, the phase speedc _r, the group velocityv _gr and the growth ratekc _i have been obtained as analytical functions of the mean zonal windU, the thermal windU _T, the coefficient of diabatic heating x, the phase differencek and the wavelengthL. Now the hypothesis is introduced, that a blocking should be expected, ifv _gr has a maximum value in the vicinity ofL _o, for whichc _r vanishes and thee-folding timet=1/kc _i (kc _i>0) is smaller than 6 days (see condition (20) in the text). One finds, that for special parameter combinations (U _T, U, ), where 15 m/secU _T25m/sec,U=10m/sec, 0.8·10^–51.5·10^–5 [sec^–1], certain valuesL _o with an appropriate phase differencek exist, which satisfy the conditions mentioned above (for values see Table 2). TherebyL _o varies within the range 8500 km <L _o<11000 km corresponding to the preferred planetary blocking wavenumber 2 in middle latitudes 50°<<70° N.     

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.     

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}      

Models of electric conductivity in the upper mantle

Summary The accessible data of the apparent resistivity _T(T) were collected and supplemented by the spectral analysis of daily means of geomagnetic components observed at world geomagnetic observatories. The data were least-squares fitted; four models of the resistivity in the upper mantle which agree with the experimental data were obtained by solving the inverse problem using Marquardt's gradient-expansion algorithm.     

Effects of collision-induced breakup on drop size distributions in steady state rainshafts

New equations and techniques for dealing with drop breakups are developed and applied to the modelling of the evolution of raindrop spectra in rainshafts. Breakup experiments byMcTaggart-Cowan andList (1975) served as data base.No matter what the original size distribution, the spectrum evolution will always lead to a Marshall-Palmer type equilibrium di tributionN=N ₀e^–D, with =constant andN ₀ proportional to the rainfall rateR. (D stands for raindrop diameter.) ForR29 mm h^–1 and an original Marshall-Palmer distribution, the required fall height to reach equilibrium is 2 km.The equilibrium distributions are characterized by linear relationships betweenR, the radar reflectivity factorZ, the liquid water content LWC and theN ₀ of the Marshall-Palmer distribution. Possible explanations for the discrepancy with observations are given.The fact that the all-water processes cannot produce drops withD2.5 mm (as confirmed by observations) leads to the conclusion that observed large raindrops withD5 mm represent melted hailstones and have not yet reached an equilibrium distribution. These latter conclusions were reached within the original assumption of videspread, steady state precipitation.     

(ϖμ/ϖT) P of the lower mantle

We estimate (/T) _P of the lower mantle at seismic frequencies using two distinct approaches by combining ambient laboratory measurements on lower mantle minerals with seismic data. In the first approach, an upper bound is estimated for |(/T) _P | by comparing the shear modulus () profile of PREM with laboratory room-temperature data of extrapolated to high pressures. The second approach employs a seismic tomography constraint ( lnV _S / lnV _P ) _P =1.8–2, which directly relates (/T) _P with (K _S /T) _P . An average (K _S /T) _P can be obtained by comparing the well-established room-temperature compression data for lower mantle minerals with theK _S profile of PREM along several possible adiabats. Both (K _S /T) and (/T) depend on silicon content [or (Mg+Fe)/Sil of the model. For various compositions, the two approaches predict rather distinct (/T) _P vs. (K _S /T) _P curves, which intersect at a composition similar to pyrolite with (/T) _P =–0.02 to –0.035 and (K _S /T) _P =–0.015 to –0.020 GPa/K. The pure perovskite model, on the other hand, yields grossly inconsistent results using the two approaches. We conclude that both vertical and lateral variations in seismic velocities are consistent with variation due to pressure, temperature, and phase transformations of a uniform composition. Additional physical properties of a pyrolite lower mantle are further predicted. Lateral temperature variations are predicted to be about 100–250 K, and the ratio of ( lnp/ lnV _S ) _P around 0.13 and 0.26. All of these parameters increase slightly with depth if the ratio of ( lnV _S / lnV _P ) _P remains constant throughout the lower mantle. These predicted values are in excellent agreement with geodynamic analyses, in which the ratios ( ln / lnV _S ) _P and ( / lnV _S ) _P are free parameters arbitrarily adjusted to fit the tomography and geoid data.     

The mean energetic level. theory

Summary One of the important atmospheric levels, the mean energetic level (MEL), which in a sense reflects the energetics of the whole atmosphere, is defined. Its fundamental properties are shown. In order to describe the MEL correctly a new vertical coordinate is introduced and discussed. The new coordinate, , is defined as the ratio of height and temperature. The MEL is shown to be a level with constant value of . Some incorrect conclusions concerning the MEL, derived in the past, have been corrected.List of symbols used c _p specific heat of air at constant pressure - c _v specific heat of air at constant volume - e base of natural logarithms - E total potential energy - f Coriolis parameter - g acceleration of gravity - i specific internal energy - I internal energy - J enthalpy - k unit vector pointing upwards - p pressure - Q diabatic heating rate - R gas constant of the air - t time - T temperature - v horizontal velocity - v ⁽³⁾ three-dimensional velocity - w vertical velocity in thez-system - z height - temperature growth rate (T/z) - Pechala's vertical coordinate (z/T) - generalized vertical velocity in the -system (d/dt) - specific potential energy - potential energy - density of the air - Ruppert function - T(1–)^–1 - ( ) _S quantity at the sea level - ( )* quantity at the MEL     

The Equatorial Ring Current as a Part of a Coupled System

The equatorial ring current is the symmetrical plasmaspheric drift current of energetic plasma trapped by the geomagnetic field. Assuming that, during a geomagnetic storm, the ring can exchange energy with other regions of the magnetosphere, a second-order differential equation must be considered as its dynamical expression, and contributions other than the solely solar-wind-induced electric field must express the forced input. The decay parameters of the recovery phase of the storm must be determined considering periods with Bz(IMF) Bz(IMF) 0.     

Experimental investigation of the elastic modulus of a fractal system—A model of fractured rocks

The elastic properties of a physical model representing a damaged rock matrix were studied using a square lattice deformed under tensile stress. The elastic modulusM of such a system varies in agreement with percolation theory as|x–x _c | ^f , wherex is the damage parameter andx _c the threshold value of the damage parameter,f3.6. Atxx _c the scale dependence ofM can be expressed asML ^–f/v , whereL is the size of the sample andv the correlation exponent in percolation theory.The experimental results are of interest in assessing elastic properties in earthquake focal zones and fault zones in general.     

Fast computation of ray synthetic seismograms in vertically inhomogeneous media

Summary A procedure of fast computation of body-wave ray synthetic seismograms in vertically inhomogeneous media is suggested. The procedure uses a special approximation of the velocitydepth distribution which guarantees continuity of the first and second derivatives of velocity and does not generate false low-velocity layers (oscillations in the velocity-depth function). The ZESY82 program package, which is based on the suggested procedure, is described. The point source with an arbitrary radiation pattern may be situated at any points of the model, the receivers are situated regularly or irregularly along any profile on the Earth's surface, containing the epicentre. Numerical examples of the synthetic record sections for a model of the Earth's crust and the uppermost mantle are given.

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¶rt;aam m¶rt; m ama ummuu a mua ¶rt;¶rt; ¶rt;, u¶rt; u nua annuauu m aa, ma nuam nm mu u n u m nu¶rt; u nu¶rt;um aau um nu mu. am nua aumn na ZESY82, a a m m¶rt;. umu aamumu uu an m ¶rt;u; nuuu ¶rt; m an¶rt; ¶rt; nu, ¶rt;a num. u¶rt; nu ummuu a ¶rt; ¶rt; ¶rt;u u amuu.

     

Normal density earth models

Summary Models of the Earth's density, close to thePREM model, have been derived, they reproduce the external normal gravitational field of the Earth and its dynamic flattening, and are referred to as normal density models. The Earth's surface is approximated by an ellipsoid of the order of the flattening, or of its square. Of the group of normal models sgtisfying the solution of the inverse problem, the normal density modelHME2 is recommended. The spherically symmetric density modelPREM, which was corrected in the course of solving the inverse problem, thus creating the modifiedPREM-E2 model, was used as the a priori information.

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¶rt; ¶rt;u an¶rt;u nmmu uu ¶rt;uPREM (m. a. a ¶rt;u nmmu), aumau n m u¶rt;mu na¶rt;am auaumau n u. m u annuum am unu¶rt; au. uau amu a ¶rt; mam H==0.003 273 994. ma ¶rt; a ¶rt; ¶rt;m ¶rt;HME2. am anu u a ¶rt; nmmu a unaa ¶rt; a¶rt;ua umua ¶rt;PREM. ¶rt;aam ¶rt;uuau m ¶rt;u n¶rt; aauPREM-E2.

     

Regional trends in European induction data

n ¶rt;a, n¶rt;mau 531 au ¶rt; u aum m u¶rt;uu n mumu ana¶rt;, ¶rt; u -m n, aauum ¶rt;um u u amuaa n¶rt;naa ma ua aama. uu nmam an¶rt;u ¶rt; u aum aam ¶rt;au cuu uP _n ^m , n¶rt;am mn n=1, 2, 3 u 5 (m n). u uua ¶rt;a¶rt;amu uu n¶rt;mauu uum au (a. 1) u u n aumam uu nmu, m n¶rt;mam u¶rt; am uuu ¶rt; u aum (u. 1–4). annuau 2 u 5 mn nm ma am mmmu m (u. 5, 6). ama uuu u m aam amu uu uma.     

Analysis and optimization of wide-band force-balance seismometer responses

¶rt; aau n¶rt;am uu, umu,au mummu u ¶rt;uau ¶rt;uanaa mu um. am n a nmua amm aamumuu um ¶rt; au uu nuu. ¶rt;ma ummuu m¶rt; nmuau mu um a a¶rt;a an¶rt;u n n¶rt;am uu n nmu.     

The optimal shape design method applied to modelling thermal thinning of the oceanic lithosphere near hot spots

Summary Using the optimal shape design method, which is generally described, and von Herzen's et al. measurements of the heat flow, the shape of the lithosphere and its thermal field is computed in the vertical plane parallel to the hot spot source versus the plate velocity at a distance of about 250 km from the axis of the Hawaiian Island chain. The results are compared with the computations based on Crough's idea of thermal rejuvenation of the oceanic lithosphere above a hot spot source. If we assume that the lateral cross-section of the lithospheric bottom is described by the Gaussian curve h=h₀ exp (–y²/2²), we obtain h₀35 km and 130 km, where h is the value of lithospheric thinning and y the lateral coordinate. We thus obtain the lower limit of the lateral dimension of the Hawaiian anomaly.

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u m¶rt; nmua nua amu, m u ma nuam, u ¶rt;a mn nm, ua a um u mn n mua nmu, naa mu um mum umua mu u ¶rt;a nuuum 250 m uuuaa aunaa. mam a uuu, au a u¶rt;u aa (Crough), aauu mn mu au um a¶rt; umu mu. u n¶rt;num, m ama nn u umu ¶rt;a nuam u aa h=h₀ exp (–y²/2²), m num h₀ 35 u 130 ,¶rt; h—umu mu u —amaa ¶rt;uama. ¶rt;am, =130 m u n¶rt; ama aaaa aauu.

     

Physical properties of the Earth's core

Calculations of the compression and temperature gradient of the core are facilitated by the use of the thermodynamic Grüneisen ratio, =3Ks/C _P . A pressure-dependent factor in is found to have the same numerical value for the core as for laboratory iron, justifying the use of a constant value for (1.6) in core calculations. The density of the outer core is satisfied by the assumption that it contains about 15% of light elements, particularly sulphur, whereas the inner core is probably ironnickel with very little lighter component. The presence of sulphur in the outer core reduces its liquidus at least 600° below pure iron, so that the adiabatic gradient does not intersect the liquidus, as Higgins and Kennedy have shown would occur in a pure iron core. The inner core is probably close to its melting point, 4700 K, and the adiabatic temperature gradient of the outer is calculated with this as a fixed point, giving 3380 K at the core-mantle boundary. The estimated electrical resistivity of the outer core, 3×10^–6 m, corresponds to a thermal conductivity of 28 W·m^–1·deg^–1, which, with the adiabatic core gradient gives a minimum of 3.9×10¹² W of heat conduction to the mantle. The only plausible source of this much heat is the radioactive decay of potassium in the core. As pointed out by Goles, Lewis, and Hall and Murthy, the presence of potassium becomes geochemically probable once sulphur is admitted as a core constituent. Thus it appears that the recognition of sulphur in the core resolves the two major difficulties which we have faced in attempting to understand the core.List of Symbols a equilibrium atomic spacing at zero pressure, also a constant - A surface area of core - b a constant - c a constant - C _V ,C _P specific heat at constant volume, constant pressure - D dimension of core (or core eddy) - E(r) atomic interaction energy - E energy due to atomic displacement from equilibrium - lattice energy of material - f ₁,f ₂ structure-dependent constants - F(P) pressure dependent factor in Grüneisen's ratio - g gravitational acceleration; also a constant (Equation (13)) - H latent heat of solidification - I integral (Equation (23)) - k Boltzmann's constant - K incompressibility (bulk modulus) - K _T ,K _S isothermal, adiabatic incompressibilities - N number of atoms in a volume of material - P pressure - dQ/dt core to mantle heat flux - r atomic spacing - r _e equilibrium value ofr under pressure - R _m magnetic Reynolds number - T temperature - T _c critical temperature - T _R reduced temperature (Equation (39)) - U specific internal energy of a material - v velocity of internal core motion - V volume - 3 volume expansion coefficient - compressibility - thermodynamic Grüneisen ratio (Equation(2)) - magnetic diffusivity - thermal conductivity - _e electronic contribution to - ₀ permeability of free space - density - _e electrical resistivity - R reduced conductivity,eM/e