Formation of cracks due to moving source over a thick crust

Summary In this paper the problem of a point source of stress moving over the surface of a thick aelotropic plate resting of a rigid foundation has been considered. Following the method ofAleksandrov & Vorovich (1960) the stress componentsZ _x andZ _z have been expanded in series of ascending powers of 1/h when the source velocity is less than (c ₄₄/)^1/2. When the velocity exceeds (c ₄₄/)^1/2 it has been shown that two cracks are produced in different directions and their successive reflections at the upper and lower surface are also obtained.     

A note on the effects of nonuniform spreading velocity of submarine slumps and slides on the near-field tsunami amplitudes

The effects of variable speeds of spreading of submarine slides and slumps on near-field tsunami amplitudes are illustrated. It is shown that kinematic models of submarine slides and slumps must consider time variations in the spreading velocities, when these velocities are less than about 2c_T, where is the long period tsunami velocity in ocean of constant depth h. For average spreading velocities greater than 2c_T, kinematic models with assumed constant spreading velocities provide good approximation for the tsunami amplitudes above the source.     

On the propagation of spherical waves due to large underground explosion

Summary When a high explosive pressure acts on the boundary of a spherical cavity in an elastic medium, the material in a region around the charge behaves plastically and thereafter an elastic wave is propagated. Not only that but the elastic plastic boundary moves also with a certain velocity which is less than the elastic wave velocitySabodash [3]²). In this paper, firstly the case when the elastic plastic boundary moves with a velocityV ₀. (V ₀<c) has been dealth with. The pressure in the elastic medium on this boundary has been assumed to be constant and equal toP ₀. Secondly the case when the elastic plastic boundary moves with an exponentially decaying velocity has been considered. The pressure in the elastic medium on this boundary has also been assumed to be of exponential character.     

A note on tsunami amplitudes above submarine slides and slumps

Tsunami generated by submarine slumps and slides are investigated in the near-field, using simple source models, which consider the effects of source finiteness and directivity. Five simple two-dimensional kinematic models of submarine slumps and slides are described mathematically as combinations of spreading constant or slopping uplift functions. Tsunami waveforms for these models are computed using linearized shallow water theory for constant water depth and transform method of solution (Laplace in time and Fourier in space). Results for tsunami waveforms and tsunami peak amplitudes are presented for selected model parameters, for a time window of the order of the source duration.The results show that, at the time when the source process is completed, for slides that spread rapidly (c_R/c_T≥20, where c_R is the velocity of predominant spreading), the displacement of the free water surface above the source resembles the displacement of the ocean floor. As the velocity of spreading approaches the long wavelength tsunami velocity the tsunami waveform has progressively larger amplitude, and higher frequency content, in the direction of slide spreading. These large amplitudes are caused by wave focusing. For velocities of spreading smaller than the tsunami long wavelength velocity, the tsunami amplitudes in the direction of source propagation become small, but the high frequency (short) waves continue to be present. The large amplification for c_R/c_T1 is a near-field phenomenon, and at distances greater than several times the source dimension, the large amplitude and short wavelength pulse becomes dispersed.A comparison of peak tsunami amplitudes for five models plotted versus L/h (where L is characteristic length of the slide and h is the water depth) shows that for similar slide dimensions the peak tsunami amplitude is essentially model independent.     

TRANSIENT FINITE INTEGRAL POINT SOURCE RESPONSE FROM A PLANAR ACOUSTIC INTERFACE: THE CAUSALITY APPROACH*

Point source reflection/transmission responses from planar interfaces based on the Sommerfeld-Weyl integral are causal and are given in the form of a solution integral with an infinite integration limit. They can be simply transformed to integrals with a finite limit by an approach that takes into account causality as well as time-symmetry aspects. To introduce the fundamental principles involved, the transformation is applied to the simple case of a point-source response from a planar acoustic interface. For a point source above a planar interface separating an upper and lower acoustic half-space of velocity c₁and c₂ the inhomogeneous plane waves in the Sommerfeld-Weyl source representation do not contribute to the final result if c₁≤c₂ and only a limited spectrum of inhomogeneous plane waves comes into play if c₁c₂.     

Transient Divergent Flow and Transport in an Infinite Anisotropic Porous Formation

When seeking to predict plume geometry resulting from fluid injection through partially penetrating wells, it is common to assume a steady-state spherically diverging flow field. In reality, the flow field is transient. The steady-flow assumption is likely to cause overestimation of injection plume radius since the accommodation of fluid by increases in porosity and fluid density is ignored. In this paper, a transient solution is developed, resulting in a nonlinear ordinary differential equation expressing plume radius as a function of time. It is shown that the problem can be fully described by one type curve. A critical time, t_c, is identified at which the percentage error of the steady-state flow solution compared to the fully dynamic problem is less than 1%. Only for large injection rates and low permeabilities, does t_c become greater than 1 h. Nevertheless, an improved approximate solution is obtained by a simple linearization procedure. The critical time, t_c for the new approximate solution is 0.3% of that required for the steady-state flow solution.     

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.     

Attenuation characteristics of Garwhal–Kumaun Himalayas from analysis of coda of local earthquakes

Coda of local earthquakes that occurred during 2006–2007 are used to study the attenuation characteristics of the Garhwal–Kumaun Himalayas. The coda attenuation characteristics are represented in terms of coda Q or Q _c . It is observed that Q _c increases with frequency. Q _c also varies with increase in lapse time of coda waves. Q _c increases up to an 85-s average lapse time. This is similar to observations around the world reported by many workers who have interpreted this as a manifestation of the fact that heterogeneity decreases with depth. However, around a 90-s average lapse time Q _c is lower than its values for lower and higher average lapse times. This is interpreted as an indication of possible presence of a fluid-filled medium or a medium having partial melts at around a 160-km depth. Q ₀, i.e., Q _c at 1 Hz, increases, and frequency parameter n decreases with increasing lapse time, barring around a 90-s lapse time. This again shows that in general, heterogeneity decreases with increasing depth. The Q ₀ and n values for smaller lapse times are similar to those for tectonically active areas. By comparing Q _c values obtained in this study with those obtained by us using the 1999 Chamoli earthquake aftershocks, it is concluded that the crust is turbid and the mantle is more transparent. However, whether the variation in Q _c values between 1999 and 2006–2007 is temporal or not cannot be definitely established from the available data set.     

Practical and robust experimental determination of c13 and Thomsen parameter δ

We analysed the complications in laboratory velocity anisotropy measurement on shales. There exist significant uncertainties in the laboratory determination of c₁₃ and Thomsen parameter δ. These uncertainties are primarily related to the velocity measurement in the oblique direction. For reliable estimation of c₁₃ and δ, it is important that genuine phase velocity or group velocity be measured with minimum uncertainty. The uncertainties can be greatly reduced if redundant oblique velocities are measured. For industrial applications, it is impractical to make multiple oblique velocity measurements on multiple core plugs. We demonstrated that it is applicable to make multiple genuine oblique group velocity measurements on a single horizontal core plug. The measurement results show that shales can be classified as a typical transversely isotropic medium. There is a coupling relation between c₄₄ and c₁₃ in determining the directional dependence of the seismic velocities. The quasi‐P‐wave or quasi‐S‐wave velocities can be approximated by three elastic parameters.     

Thermal and magnetic instabilities in a rapidly rotating fluid sphere

Abstract

A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r ₀, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis. The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere. We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance.

Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number R_c. The critical temperature gradient is smallest (R_c reaches a minimum) when the magnetic field strength parameter Λ ? 1. [R_c and Λ are defined in (2.3).] The instability takes the form of a very slow wave with frequency of order κ/r ² ₀ and its direction of propagation changes from eastward to westward as Λ increases through Λ _c ? 4.

When the fluid is sufficiently stably stratified and when Λ > Λ_m ? 22 a new mode of instability sets in. It is magnetically driven but requires some stratification before the energy stored in the magnetic field can be released. The instability takes the form of an eastward propagating wave with azimuthal wavenumber m = 1.     

Resonance capacity method for earthquake response analysis of hysteretic structures

The Resonance Capacity Method is proposed for the earthquake response analysis of hysteretic structures. Resonance Capacity is a physical quantity of structures which is related to the hysteretic energy absorbed by structures in one cycle and is equated to the acceleration, velocity and displacement amplitudes α₀, d₀ and d₀ of earthquake ground motions at resonance.¹ According to the idealized trapezoidal approximation of earthquake ground motions in the logarithmic period–velocity plane as proposed by Veletsos and Newmark,⁸ the Resonance Capacity property applies in each period range, short, medium and long, where α₀, v₀ and d₀ respectively are approximately constant. In the medium range of periods, the energy dissipated in hysteretic loops and the deformation amplitudes of a single-degree system with elasto–plastic force–deformation relationships are calculated for the case of El Centro 1940, 18 May earthquake, by this Resonance Capacity Method. The result is compared with results from conventional numerical response analyses obtained by Berg and Thomaides,¹⁴ Kato and Akiyama¹² and Veletsos and Newmark,⁸ and the general agreement is seen to be good. Therefore, it may be possible to apply this Resonance Capacity Method over the entire range of periods. By means of this method the earthquake response analysis of hysteretic systems can be performed easily, and the hysteretic energy and fatigue characteristics of structures may be taken into account directly, up to the point of fracture.     

Surface waves in an elastic semi-space with a stable nonhomogeneity

Summary Surface waves in an isotropic nonhomogeneous elastic semi-space (plane strain) are studied. It is assumed that the Poisson's ratio and the density of the medium are constant but shear modulus is a monotonic function of the depth. Use of the stress equation of motion is made to reduce the problem to an eigenvalue problem for a differential equation of fourth order with polynomial coefficients. A series solution of the problem is obtained and dependence of the Rayleigh velocityC _R on the wavelength and the nonhomogeneity of the medium, is studied. Group velocity period curves are also obtained.

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Zusammenfassung Oberflächenwellen in einem isotropen Nicht-homogenen elastischen Halbraum (Für den ebenen Verzerungszustand) werden untersucht. Die Poissonsche Querzahl und die Dichte werden als konstant vorausgesetzt während der Schubmodul ein monotone Funktion der Tiefe ist. Unter Benutzung der Spannungstensorsbewegungsgleichungen reduziert sich das problem zu einem Eigenwertproblem einer Differentialgleichung der vierten Ordnung mit Polynomkoeffizienten. Eine Reihenentwicklung wird als Lösung des problems erhalten und die Abhängigkeit der RayleighgeschwindigkeitC _R von der Wellenlänge und die Nicht-homogeneität des mediums werden untersucht. Ausserdem werden Gruppengeschwindigkeit und Perioden kurven erhalten.

     

A simple parameterization of bulk canopy resistance from climatic variables for estimating hourly evapotranspiration

This paper examines a model for estimating canopy resistance r_c and reference evapotranspiration ET_o on an hourly basis. The experimental data refer to grass at two sites in Spain with semiarid and windy conditions in a typical Mediterranean climate. Measured hourly ET_o values were obtained over grass during a 4 year period between 1997 and 2000 using a weighing lysimeter (Zaragoza, northeastern Spain) and an eddy covariance system (Córdoba, southern Spain). The present model is based on the Penman–Monteith (PM) approach, but incorporates a variable canopy resistance r_c as an empirical function of the square root of a climatic resistance r* that depends on climatic variables. Values for the variable r_c were also computed according to two other approaches: with the r_c variable as a straight‐line function of r* (Katerji and Perrier, 1983, Agronomie 3 (6): 513–521) and as a mechanistic function of weather variables as proposed by Todorovic (1999, Journal of Irrigation and Drainage Engineering, ASCE 125 (5): 235–245). In the proposed model, the results show that r_c/r_a (where r_a is the aerodynamic resistance) presents a dependence on the square root of r*/r_a, as the best approach with empirically derived global parameters. When estimating hourly ET_o values, we compared the performance of the PM equation using those estimated variable r_c values with the PM equation as proposed by the Food and Agriculture Organization, with a constant r_c = 70 s m^?1. The results confirmed the relative robustness of the PM method with constant r_c, but also revealed a tendency to underestimate the measured values when ET_o is high. Under the semiarid conditions of the two experimental sites, slightly better estimates of ET_o were obtained when an estimated variable r_c was used. Although the improvement was limited, the best estimates were provided by the Todorovic and the proposed methods. The proposed approach for r_c as a function of the square root of r* may be considered as an alternative for modelling r_c, since the results suggest that the global coefficients of this locally calibrated relationship might be generalized to other climatic regions. It may also be useful to incorporate the effects of variable canopy resistances into other climatic and hydrological models. Copyright © 2005 John Wiley & Sons, Ltd.     

Statistical Detection and Characterization of a Deviation from the Gutenberg-Richter Distribution above Magnitude 8

— We present a quantitative statistical test for the presence of a crossover c₀ in the Gutenberg-Richter distribution of earthquake seismic moments, separating the usual power-law regime for seismic moments less than c₀ from another faster decaying regime beyond c₀. Our method is based on the transformation of the ordered sample of seismic moments into a series with uniform distribution under condition of no crossover. A simulation method allows us to estimate the statistical significance of the null hypothesis H₀ of an absence of crossover (c₀=infinity). When H₀ is rejected, we estimate the crossover c₀ using two different competing models for the second regime beyond c₀ and the simulation method. For the catalog obtained by aggregating 14 subduction zones of the Circum-Pacific Seismic Belt, our estimate of the crossover point is log(c₀)=28.14 ± 0.40 (c₀ in dyne-cm), corresponding to a crossover magnitude m_W=8.1 ± 0.3. For separate subduction zones, the corresponding estimates are substantially more uncertain, so that the null hypothesis of an identical crossover for all subduction zones cannot be rejected. Such a large value of the crossover magnitude makes it difficult to associate it directly with a seismogenic thickness as proposed by many different authors. Our measure of c₀ may substantiate the concept that the localization of strong shear deformation could propagate significantly in the lower crust and upper mantle, thus increasing the effective size beyond which one should expect a change of regime.     

Some exact solutions for the flow of fluid through tubes with uniformly porous walls

Summary This paper considers an incompressible fluid flowing through a straight, circular tube whose walls are uniformly porous. The flow is steady and one dimensional. The loss of fluid through the wall is proportional to the mean static pressure in the tube. Several formulations of the wall shear stress are considered; these formulations were motivated by the results from Hamel's radial flow problem, boundary layer flows/and boundary layer suction profiles. For each of these formulations exact solutions for the mean axial velocity and the mean static pressure of the fluid are obtained. Sample results are plotted on graphs. For the constant wall shear stress problem, the theoretical solutions compare favorably with some experimental results.Notations A, B, D, E constant parameters - a, b constant parameters - Ai(z), Bi(z) Airy functions - Ai, Bi derivatives of Airy functions - k constant of proportionality betweenV andp - L length of pores - p,p mean static pressure - p ₀ static pressure outside the tube - p ₀ value ofp atx=0 - Q constant exponent - R inside radius of the tube - T wall shear stress - T ₀ shear parameter - t wall thickness - U free stream velocity - ,u mean axial velocity - u ₀ value ofu atx=0 - V,V mean seepage velocity through the wall - v ₀ mean seepage velocity - x,x axial distance along the tube - z transformed axial distance - z ₀ value ofz atx=0 - mean outflow angle through the wall - cos - density of the fluid - wall shear stress - dynamic viscosity of the fluid - over-bar dimensional terms - no bar nondimensional terms The National Center for Atmospheric Research is sponsored by the National Science Foundation     

Inundation flow velocity of tsunami on land

An estimation of tsunami inundation flow velocity is one of the most challenging issues among tsunami research. Based on field data of inundation depth and inundation flow velocity u estimated using Bernoulli's theorem and inundation depth, fundamental characteristics of the relationship between inundation flow velocity and inundation depth are examined. Fundamental characteristics of the velocity coefficient where g is gravitational acceleration, h_f and h_r are inundation depths at the front and the back of structures such as a rectangular building with vertical walls, respectively) implicitly included in the relationship are examined through hydraulic experiments. As a result, C_v = 0.6 is recommended as its simple and practical value. It is confirmed through these examinations that the Froude number, defined by where , ranges 0.7–2.0, and when C_v = 0.6 is adopted this Froude number ranges 0.42–1.2. By using the relationship and C_v = 0.6, two simple and practical relationships are presented for two cases where inundation flow velocity exerts the largest or the smallest fluid force on structures. These relationships can be used to roughly grasp the practical side of tsunami damage, and estimate fluid force acting on individual structures, moving velocity and collision force of floating objects and sediment transport such as boulder and sand. Fundamental characteristics of the waterline (tsunami trace) distribution around/on the typical object model (square pillar, corn and column) are also examined through steady flow experiments, and it is confirmed that the maximum and the minimum values of h_f/h₀ in the full type model of the square pillar are almost the same as those of h_f/h_r obtained by field surveys where h₀ is uniform flow depth. It is also confirmed that h_r ? h₀ when the Froude number, defined by where u₀ is uniform flow velocity, is much less than 1.0. Using a newly defined velocity coefficient, tsunami inundation flow velocity on land can be estimated practically and would be useful for checking proposed sediment transport models that are now being developed by tsunami geologists.     

Attenuation of coda waves and Q c value beneath the Chengdu telemetered seismic network

Based on the single scattering model of coda power spectrum analysis, digital waveform data of 50 events recorded by the real-time processing system of the Chengdu telemetry network are analyzed to estimate the Q _c values of earth medium beneath the Chengdu telemetry network for several specified frequencies. It is found that the Q _c shows the frequency dependency in the form of Q _c = Q ₀ f ⁿ in the range of 1.0 to 20.0Hz. Estimated Q ₀ ranges from 60.83 to 178.05, and n is found to be 0.713 to 1.159. The average value of Q ₀ and n are 117 and 0.978 respectively. This result indicates the strong frequency dependency of the attenuation of coda waves beneath the Chengdu telemetry network. Comparing with the results obtained in other regions of the world, it is found that Q ₀ ⁻¹ value and its change with frequency are similar to those in regions with strong tectonic activity. This subject is supported by the Ministry of Personnel, China for partly sponsoring.     

Lapse time and frequency-dependent attenuation characteristics of coda waves in the Northwestern Himalayas

We analyzed the local earthquakes waveform recorded on a broadband seismic network in the northwestern Himalayan Region to compute lapse time and frequency dependence of coda Q (Q _c). The observed Q _c values increase with increasing lapse time at all frequency bands. The increase in Q _c values with lapse time is attributed to an increase in Q _c with depth. This implies that attenuation decreases with increasing depth. The approximate radius of medium contributing to coda generation varies from 55 to 130 km. By comparing the Q _c values with those from other regions of the world, we find that they are similar to those obtained from tectonically active regions. The estimated Q _c values show a frequency-dependent relationship, Q _c = Q ₀ f ⁿ , where Q ₀ is Q _c at 1 Hz and n represents degree of frequency dependence. They represent the level of heterogeneity and tectonic activity in an area. Our results show that northwest Himalayas are highly heterogeneous and tectonically very active. Q ₀ increases from 113 ± 7 to 243 ± 10 and n decreases from 1.01 ± 0.05 to 0.85 ± 0.03 when lapse time increases from 30 to 70 s. As larger time window sees the effect of deeper part of the Earth, it is concluded that Q ₀ increases and n decreases with increasing depth; i.e., heterogeneity decreases with depth in the study area.     

Friction constitutive law with rate and state dependences

A general formula for the Dieterich-Ruina friction constitutive law with rate and state (n-state variables,n=1, 2,...) dependences has been obtained and discussed under the assumption that the slip acceleration a varies ion a linearly with the slip displacement , namelya = a ₀ + (-₀). Wherea ₀, ₀ are initial constants, is the acceleration rate and constant.a ₀ and may be arbitrary constants (positive, negative or zero).The extreme value of frictional resistance and the existence condition of the extreme value, which are very important and govern to some degree the motion process of a frictionally slipping mechanical system, have been analyzed. A critical value _c which is the measure of the velocity weakening and velocity strengthening of the mechanical system, and its properties and the relationship to the extreme problem have been studied. Again, according to the critical value _c, the concepts of light or strong velocity weakening (or strengthening) are introduced.A possibly new phenomenon that frictional resistance may vary in some kind of decayed oscillation is found. Finally, the condition for the smallest frictional resistance for a slipping mechanical system with nonuniform acceleration has been obtained.     

TWO-AND-ONE-HALF DIMENSIONAL IN-PLANE WAVE PROPAGATION*

This paper collects certain results concerning wave propagation in two-and-one-half dimensions, i.e., three-dimensional (3-D) wave propagation in a medium that has variations in two dimensions only. The results of interest are for sources and receivers in the plane determined by the two directions of parameter variation. The objective of this work is to reduce the analysis of the in-plane propagation to 2-D analysis while retaining–at least asymptotically–the proper 3-D geometrical spreading. We do this for the free space Green's function and for the Kirchhoff approximate upward scattered field from a single reflector. In both cases the derivation is carried out under the assumption of a background velocity c(x, z) with the special cases c = c₀ and c = c(z).