Electromagnetic induction in the earth due to stationary and moving sources

A new approach to the theory of electromagnetic induction is developed that is applicable to moving as well as stationary sources. The source field is considered to be a standing wave generated by two waves travelling in opposite directions along the surface of the earth. For a stationary source the incident waves have velocities of the same magnitude, however for a moving source the velocities of the two incident waves are respectively increased and decreased by the velocity of the source. Electromagnetic induction in the earth is then considered as refraction of these waves and gives, for both stationary and moving sources, the magnetotelluric relation: $$\frac{{ - E_y }}{{H_x }} = \left( {\frac{{i\omega \mu }}{\sigma }} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - i\frac{{v^2 }}{{\omega \mu \sigma }}} \right)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where ν is the wavenumber of the source, μ is the permeability (4π·10^?7) and σ is the conductivity of the earth. ω is the angular frequency of the variation observed on the earth. For a stationary source the observed frequency is the same as the source frequency, however the effect of moving a time-varying source is to make the observed frequency different from the frequency of the source. Failure to recognise this in previous studies led to some erroneous conclusions. This study shows that a moving source isnot “electromagnetically broader” than a stationary source as had been suggested.     

A theoretical value for von Karman's constant

This paper extends the theory of the entity and entrainment model of turbulence to obtain a numerical value of von Karman's constant,k=0.37. The formula is, $$k = (2a^3 /A)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \ln \beta $$ where,a=1/12 is the entrainment constant,A=1 is the turbulent decay constant, and β is the ratio in height of the successive self-similar layers of the theory, where β is evaluated as β=e ². These new values fork and β improve the surface roughness length estimates derived from this theory.     

Edge Detection and Depth Estimation Using a Tilt Angle Map from Gravity Gradient Data of the Kozaklı-Central Anatolia Region,Turkey

In this paper the application of an edge detection technique to gravity data is described. The technique is based on the tilt angle map (TAM) obtained from the first vertical gradient of a gravity anomaly. The zero contours of the tilt angle correspond to the boundaries of geologic discontinuities and are used to detect the linear features in gravity data. I also present that the distance between zero and ±p\mathord

Estimates of fracture parameters of earthquakes

A new estimate of the fracture parameters of earthquakes is provided in this paper. By theMuskhelishvili method (1953) a number of basic relations among fracture-mechanics parameters are derived. A scheme is proposed to evaluate the slip weakening parameters in terms of fault dimension, average slip, and rise time, and the new results are applied to 49 events compiled in the earthquake catalogue ofPurcaru andBerckhemer (1982). The following empirical relations are found in the paper: $$\begin{gathered} \frac{{\tau _B - \tau _f }}{{\tau _\infty - \tau _f }} = 2.339 \hfill \\ {{\omega _c } \mathord{\left/ {\vphantom {{\omega _c } {W = 0.113}}} \right. \kern-\nulldelimiterspace} {W = 0.113}} \hfill \\ \log G_c \left( {{{dyne} \mathord{\left/ {\vphantom {{dyne} {cm}}} \right. \kern-\nulldelimiterspace} {cm}}} \right) = 2 \log L (km) + 6.167 \hfill \\ \log \delta _c (cm) = 2 \log L (km) - 1.652 \hfill \\ \end{gathered} $$ whereG _c is the specific fracture energy,ω _c the size of the slip weakening zone,δ _c the slip weakening displacement,τ _B ?τ _f the drop in strength in the slip weakening zone,τ _∞?τ _f the stress drop,L the fault length, andW the fault width. The investigation of 49 shocks shows that the range of strength dropτ _B ?τ _f is from several doze to several hundred bars at depthh<400 km, but it can be more than 10³ bars ath>500 km; besides, the range of the sizeω _c of the strength degradation zone is from a few tenths of a kilometer to several dozen kilometers, and the range of the slip weakening displacementδ _c is from several to several hundred centimeters. The specific fracture energyG _c is of the order of 10⁸ to 10¹¹ erg cm^?2 when the momentM ₀ is of the order of 10²³ to 10²⁹ dyne cm.     

The viscosity of hydrous dacitic liquids: implications for the rheology of evolving silicic magmas

The viscosity of a series of six synthetic dacitic liquids, containing up to 5.04 wt% dissolved water, was measured above the glass transition range by parallel-plate viscometry. The temperature of the 10¹¹ Pa s isokom decreases from 1065 K for the anhydrous liquid, to 864 K and 680 K for water contents of 0.97 and 5.04 wt% H₂O. Including additional measurements at high temperatures by concentric-cylinder and falling-sphere viscometry, the viscosity (η) can be expressed as a function of temperature and water content w according to: where η is in Pa s, T is temperature in K, and w is in weight percent. Within the conditions of measurement, this parameterization reproduces the 76 viscosity data with a root-mean square deviation (RMSD) of 0.16 log units in viscosity, or 7.8 K in temperature. The measurements show that water decreases the viscosity of the dacitic liquids more than for andesitic liquids, but less than for rhyolites. At low temperatures and high water contents, andesitic liquids are more viscous than the dacitic liquids, which are in turn more viscous than rhyolitic liquids, reversing the trend seen for high temperatures and low water contents. This suggests that the relative viscosity of different melts depends on temperature and water content as much as on bulk melt composition and structure. At magmatic temperatures, rhyolites are orders of magnitude more viscous than dacites, which are slightly more viscous than andesites. During degassing, all three liquids undergo a rapid viscosity increase at low water contents, and both dacitic and andesitic liquids will degas more efficiently than rhyolitic liquids. During cooling and differentiation, changing melt chemistry, decreasing temperature and increasing crystal content all lead to increases in the viscosity of magma (melt plus crystals). Under closed system conditions, where melt water content can increase during crystallization, viscosity increases may be small. Conversely, viscosity increases are very abrupt during ascent and degassing-induced crystallization.     

Die Bestimmung einer Geoiderhebung aus Messungen mit der Drehwaage von Eötvös

Zusammenfassung Aus Drehwaagenmessungen des Bundesamtes für Eich- und Vermessungegswesen wird in einem Spezialfall die Geoidhebung eines fast isolierten Gebirgsstockes durch zweimalige Integration von bestimmt. Direkte Berechnung des Potentials führt zum gleichen Resultat.

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Résumé Au moyen de mesures faites avec la balance de torsion d'Eötvös effectuées par le Service Fédéral Autrichien de Géodésie, de Cartographie, du Cadastre et des Poids et Mesures, un cas particulier d'élévation du géoide produite par un massif de montagne presque isolé est déterminé par intégration réitérée de . Le calcul direct du potentiel a conduit au même résultat.

     

Electric conductivities of a cracked solid

Calculations on the basis of the self-consistent approximation are used to study the effects of randomly distributed elliptical cracks and of non-randomly distributed circular cracks, either dry or saturated by a highly conductive material phase, on the electric conductivities of a cracked body. Analytic and numeric results are given for two special non-random distributions. In the first, the cracks are assumed randomly distributed in planes parallel to a given plane. In the second, the crack normals are randomly distributed in parallel planes. The results of the theoretical calculations indicate that the magnitudes of the crack induced variations of the dry cracked rock depend upon a crack density parameter ? rather than upon the crack porosity. Here, ? is defined as $$\varepsilon = \frac{{2N}}{\pi }< \frac{{A^2 }}{P} > $$ whereN is the average number of cracks per unit volume, andA andP are the crack area and perimeter respectively. (For circular cracks of radiusa, ?=N〈a³〉.) Although a straightforward relationship does connect ? with the porosity, it may be more meaningful for laboratory experiments to concentrate upon measuring crack-induced variations as functions of crack density rather than of porosity. For saturated cracked rocks, the results of the calculations indicate that, in addition to ?, variations in conductivity depend also upon a saturation parameter Ω, which relates crack aspect ratio α to matrix and fluid conductivities σ and σ_F $$\Omega = \frac{{{\sigma \mathord{\left/ {\vphantom {\sigma {\sigma _F }}} \right. \kern-\nulldelimiterspace} {\sigma _F }}}}{\alpha }.$$      

Flow velocity profiles in the Lower Scheldt estuary

Recent acoustic Doppler current profiler (ADCP)-measurements in the Scheldt estuary near Antwerp, Belgium, revealed anomalous, i.e. anti-clockwise circulations in a left bend during the major part of the flood period; these circulations were established shortly after the turn of the tide. During ebb, anti-clockwise circulations persisted, as predicted by classical theory. These data were analysed with a 3D and a 1DV-model. The 3D simulations reveal that the anomalous circulations are found when salinity is included in the computations—without salinity “normal” circulations were found. From analytical and 1DV simulations, it is concluded that a longitudinal salinity gradient ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ may induce a near-bed maximum in flow velocity reversing the direction of the secondary currents. The 1DV-model was then used to assess the contribution of various processes one by one. It was found that because of a reduction in vertical mixing, the vertical velocity profile is not at equilibrium during the first phase of accelerating tide, further enhancing the effects of ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ . A small vertical salinity gradient ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial z}}} \right. \kern-0em} {\partial z}$ appeared to have a very large effect as the crosscurrents of the secondary circulations induced by ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ became an order of larger magnitude. However, at the site under consideration, the effects of transverse salinity gradients, generated by differential advection in the river bend, were dominant: adverse directions of the secondary circulations were found even when the vertical velocity profile became more regular with a more or less logarithmic shape, i.e. when the effects of ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial x}}} \right. \kern-0em} {\partial x}$ and ${\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial z}}} \right. \kern-0em} {\partial z}$ did not play a dominant role anymore. It is argued that data on the secondary velocity structure, which can be measured easily owing to today’s developments in ADCP equipment, may serve as an indicator for the accuracy at which the salinity field is computed with 3D numerical models. Moreover, the large effect of the salinity structure on the velocity field must have a large impact on the morphological development of estuaries, which should therefore be accounted for in morphological modelling studies.     

State of stress in the lithosphere: Inferences from the flow laws of olivine

The experimental flow data for rocks and minerals are reviewed and found to fit a law of the form $$\dot \varepsilon = A'\left[ {sinh (\alpha \sigma )} \right]^n \exp \left[ {{{ - (E * + PV * )} \mathord{\left/ {\vphantom {{ - (E * + PV * )} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} \right]$$ where \(\dot \varepsilon \) This law reduces to the familiar power-law stress dependency at low stress and to an exponential stress dependency at high stress. Using the material flow law parameters for olivine, stress profiles with depth and strain rate are computed for a representative range of temperature distributions in the lithosphere. The results show that the upper 15 to 25 km of the oceanic lithosphere must behave elastically or fail by fracture and that the remainder deforms by exponential law flow at intermediate depths and by power-law flow in the rest. A model computation of the gravitational sliding of a lithospheric plate using olivine rheology exhibits a very sharp decoupling zone which is a consequence of the combined effects of increasing stress and temperature on the flow law, which is a very sensitive function of both.     

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.     

Attenuation of P,S, and coda waves in Koyna region,India

The attenuation properties of the crust in the Koyna region of the Indian shield have been investigated using 164 seismograms from 37 local earthquakes that occurred in the region. The extended coda normalization method has been used to estimate the quality factors for P waves and S waves , and the single back-scattering model has been used to determine the quality factor for coda waves (Q _c). The earthquakes used in the present study have the focal depth in the range of 1–9 km, and the epicentral distance vary from 11 to 55 km. The values of and Q _c show a dependence on frequency in the Koyna region. The average frequency dependent relationships (Q = Q ₀ f ⁿ) estimated for the region are , and . The ratio is found to be greater than one for the frequency range considered here (1.5–18 Hz). This ratio, along with the frequency dependence of quality factors, indicates that scattering is an important factor contributing to the attenuation of body waves in the region. A comparison of Q _c and in the present study shows that for frequencies below 4 Hz and for the frequencies greater than 4 Hz. This may be due to the multiple scattering effect of the medium. The outcome of this study is expected to be useful for the estimation of source parameters and near-source simulation of earthquake ground motion, which in turn are required in the seismic hazard assessment of a region.     

Zur Kinematik der magnetischen Feldlinien im Plasma

Zusammenfassung Die Kinematik der magnetischen Feldlinien im Plasma kann mit denselben mathematischen Hilfsmitteln studiert werden, welche sich in der Kinematik der Wirbel bewährt haben. Ausgehend vom Faradayschen Induktionsgesetz für bewegte Medien können gefolgert werden: eine notwenige und hinreichende bedingung dafür, dass die magnetischen Feldlinien mit materiellen Kurven zusammenfallen; ein Analogon zuC. Truesdells «basic vorticity formula», welches die Mitführung und Diffusion der magnetischen Feldlinien im Plasma beschreibt; Sätze zur Kinematik der Feldlinien, welche eine frei wählbare tensorielle Feldfunktion beliebiger Stufe enthalten und den vonH. Ertel formulierten «allgemeinen Wirbelsätzen» entsprechen, insbesondere Analoga zuErtels «Vertauschungsrelationen». In einem isentropen idealen Plasma ist das mit dem spezifischen Volumen multiplizierte Skalar-produkt aus der magnetischen Induktion und dem Gradienten der Entropiedichte zeitlich individuell konstant.

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Summary The kinematics of magnetic field lines in a plasma can be studies by means of the mathematical methods used in the kinematics of vorticity. Starting withFaraday's law of induction for moving circuits the following results can be derived: a necessary and sufficient condition that the magnetic field lines remain material lines; a formula describing the convection and diffusion of the magnetic field lines in a plasma, which is analogous to the «basic vorticity formula» ofC. Truesdell; general theorems containing an arbitrary tensor field of any order, which are analogous to general vorticity theorems ofH. Ertel, especially a «commutation formula» corresponding to the «Euler-Ertel commutation formula» for circulation preserving motions. Given an isentropic ideal plasma it follows that ( denoting the density, the magnetic induction,s the specific entropy, andd/dt the material time derivative).

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Herrn ProfessorDr. Hans Ertel zum 60. Geburtstag in Dankbarkeit gewidmet.     

The boundary value problem of Poisson’s equation with Stokes’ boundary condition

The following Poisson’s equation with the Stokes’ boundary condition is dealt with $$\left\{ \begin{gathered} \nabla ^2 T = - 4\pi Gp outside S, \hfill \\ \left. {\frac{{\partial T}}{{\partial h}} = \frac{1}{\gamma }\frac{{\partial y}}{{\partial h}}T} \right|_s = - \Delta g, \hfill \\ T = O\left( {r^{ - 3} } \right) at infinity, \hfill \\ \end{gathered} \right.$$ whereS is reference ellipsord. Under spherical approximation transformation, the ellipsoidal correction terms about the boundary condition, the equation and the density in the above BVP are respectively given. Therefore, the disturbing potentialT can he obtained if the magnitudes aboveO(ε⁴) are neglected.     

Estimation of coda wave attenuation in East Central Iran

The attenuation of coda waves, Q _c , has been estimated in Zarand, Jiroft, and Bam regions of east central Iran using a single back-scattering model of S-coda envelopes. For this purpose, the recordings of 97 earthquakes by three seismic networks and a local strong ground motion network have been used. In this research, the frequency-dependent Q _c values are estimated at central frequencies of 1.5, 3, 6, 8, 12, 16, and 24 Hz using different lapse time windows from 20 to 60 s. The frequency-dependent relationships obtained are for Zarand, for Jiroft, and for Bam region. From the strong ground motion data, we obtain the relation . The Q _c frequency-dependent relationship for the entire region of east central Iran from all data (both seismograms and accelerograms) is . The average Q _c values estimated and their frequency dependent relationships correlate well with a highly heterogeneous and highly tectonically active region. Results also show that the attenuation is higher in Bam region compared to Zarand and Jiroft regions.     

A semiempirical mathematical model of the secondary pollution of water bodies by soluble iron and manganese forms

A semiempirical mathematical model of iron and manganese migration from bottom sediments into the water mass of water bodies has been proposed based on some basic regularities in the geochemistry of those elements. The entry of dissolved forms of iron and manganese under aeration conditions is assumed negligible. When dissolved-oxygen concentration is <0.5 mg/L, the elements start releasing from bottom sediments, their release rate reaching its maximum under anoxic conditions. The fluxes of dissolved iron and manganese (Me) from bottom sediments into the water mass (J _Me) are governed by the gradients of their concentrations in diffusion water sublayer adjacent to sediment surface and having an average thickness of h = 0.025 cm: \({J_{Me}} = - {D_{Me}}\frac{{{C_{Me\left( {ss} \right)}} - {C_{Me\left( w \right)}}}}{h}\) (D _Me ≈ 1 × 10^–9 m²/s is molecular diffusion coefficient of component Me in solution; C _Me(ss) and C _Me(w) ≈ 0 are Me concentrations on sediment surface, i.e., on the bottom boundary of the diffusion water sublayer, and in the water mass, i.e., on the upper boundary of the diffusion water sublayer). The value of depends on water saturation with dissolved oxygen (\({\eta _{{O_2}}}\)) in accordance with the empiric relationship \({C_{Me\left( {ss} \right)}} = \frac{{C_{_{Me\left( {ss} \right)}}^{\max }}}{{1 + k{\eta _{{O_2}}}}}\) (k is a constant factor equal to 300 for iron and 100 for manganese; C _Me(ss) ^max is the maximal concentration of Me on the bottom boundary of the diffusion water sublayer with C _Fe(ss) ^max ≈ 200 μM (11 mg/L), and C _Mn(ss) ^max ≈ 100 μM (5.5 mg/L).     

Effects of compressibility on the flow of lava

Lava contains gas bubbles and hence is a compressible liquid whose density increases as a function of pressure. After eruption at the Earth's surface, it spreads at a rate which is a function of its thickness and it is compressed under its own weight. Therefore, both thickness and spreading rate are determined by a balance between viscous and compressible effects. Theoretical equations are derived for the shape and velocity of a compressible liquid spreading on a horizontal surface. Solutions are obtained for a fixed eruption rate Q. The radial extent of the flow increases proportional to t ^1/2. A dimensionless number C is defined which characterizes the importance of flow compression: , where ₀ is bubbly lava density at atmospheric pressure, compressibility, viscosity and g the acceleration of gravity. C can be thought of as the ratio of two characteristic length-scales, one for compression effects and one for viscous effects. The larger C is, the more important compressibility effects are. As C is increased, the flow becomes thinner because the liquid is compressed more and more efficiently. Compressibility acts to smooth out variations of flow thickness, which provide the driving force. Thus, all else being equal, a compressible liquid flows less rapidly than an incompressible one. When trying to infer the effective viscosity of a flow from its spreading rate, the neglect of compressibility leads to an overestimate. The various factors which act to determine the distribution of gas bubbles in lava flows are reviewed and discussed quantitatively. Comparison with data from Obsidian Dome (Eastern California) shows that disequilibrium effects are important and that bubble resorption during burial in a thick flow is not a pervasive phenomenon. The analysis is applied to the 1979 dome of Soufrière de Saint Vincent (W.I.). An effective value of compressibility for this 100-m-thick dome is 1.5x10^–6 Pa^–1. This implies that, all else being equal, the viscosity of this lava may be overestimated by a factor of 5 if no account is taken of the compressible nature of the flow.     

The TKE dissipation rate in the northern South China Sea

The microstructure measurements taken during the summer seasons of 2009 and 2010 in the northern South China Sea (between 18°N and 22.5°N, and from the Luzon Strait to the eastern shelf of China) were used to estimate the averaged dissipation rate in the upper pycnocline 〈ε _p〉 of the deep basin and on the shelf. Linear correlation between 〈ε _p〉 and the estimates of available potential energy of internal waves, which was found for this data set, indicates an impact of energetic internal waves on spatial structure and temporal variability of 〈ε _p〉. On the shelf stations, the bottom boundary layer depth-integrated dissipation $ {\widehat{\varepsilon}}_{\mathrm{BBL}} $ reaches 17–19 mW/m², dominating the dissipation in the water column below the surface layer. In the pycnocline, the integrated dissipation $ {\widehat{\varepsilon}}_{\mathrm{p}} $ was mostly ～10–30 % of $ {\widehat{\varepsilon}}_{\mathrm{BBL}} $ . A weak dependence of bin-averaged dissipation $ \overline{\varepsilon} $ on the Richardson number was noted, according to $ \overline{\varepsilon}={\varepsilon}_0+\frac{\varepsilon_{\mathrm{m}}}{{\left(1+ Ri/R{i}_{\mathrm{cr}}\right)}^{1/2}} $ , where ε ₀ + ε _m is the background value of $ \overline{\varepsilon} $ for weak stratification and Ri _cr?=?0.25, pointing to the combined effects of shear instability of small-scale motions and the influence of larger-scale low frequency internal waves. The latter broadly agrees with the MacKinnon–Gregg scaling for internal-wave-induced turbulence dissipation.     

The seismic wave absorption in the crust and upper mantle in the vicinity of the Kislovodsk seismic station

The Q-factor estimates of the Earth’s crust and upper mantle as the functions of frequency (Q(f)) are obtained for the seismic S-waves at frequencies up to ~35 Hz. The estimates are based on the data for ~40 earthquakes recorded by the Kislovodsk seismic station since 2000. The magnitudes of these events are M_W > 3.8, the sources are located in the depth interval from 1 to 165 km, and the epicentral distances range from ~100 to 300 km. The Q-factor estimates are obtained by the methods developed by Aki and Rautian et al., which employ the suppression of the effects of the source radiation spectrum and local site responses in the S-wave spectra by the coda waves measured at a fixed lapse time (time from the first arrival). The radiation pattern effects are cancelled by averaging over many events whose sources are distributed in a wide azimuthal sector centered at the receiving site. The geometrical spreading was specified in the form of a piecewise-continuous function of distance which behaves as 1/R at the distances from 1 to 50 km from the source, has a plateau at 1/50 in the interval from 50–70 km to 130–150 km, and decays as \({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt R }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sqrt R }$}}\) beyond 130–150 km. For this geometrical spreading model and some of its modifications, the following Q-factor estimates are obtained: Q(f) ~ 85f^0.9 at the frequencies ranging from ~1 to 20 Hz and Q(f) ~ 75f^1.0 at the frequencies ranging from ~1 to 35 Hz.