Lamb's problem and comments on the paper ‘on leaking modes’ byUsha Gupta

Summary In a recent paper,Gupta [5]²⁾ re-examined the significance of leaking modes in Lamb's problem (Lamb [7]). In this paper, we present a brief review of the exact Cagniard-de Hoop solution to this problem, and use these results to examine the question of the leaking mode in more detail. The leaking mode may either cause a separate arrival,P, or influence the shape of other arrivals e.g.SpS. We have attempted to clarify and extend previous results and correct misconceptions which have appeared elsewhere and, therefore, most of this discussion is tutorial in nature.     

Strong fast long-period waves in the Efpalio 2010 earthquake records: explanation in terms of leaking modes

The January 18, 2010, shallow earthquake in the Corinth Gulf, Greece (M _w  5.3) generated unusually strong long-period waves (periods 4–8 s) between the P and S wave arrival. These periods, being significantly longer than the source duration, indicated a structural effect. The waves were observed in epicentral distances 40–250 km and were significant on radial and vertical component. None of existing velocity models of the studied region provided explanation of the waves. By inverting complete waveforms, we obtained an 1-D crustal model explaining the observation. The most significant feature of the best-fitting model (as well as the whole suite of models almost equally well fitting the waveforms) is a strong velocity step at depth about 4 km. In the obtained velocity model, the fast long-period wave was modeled by modal summation and identified as a superposition of several leaking modes. In this sense, the wave is qualitatively similar to P long or Pnl waves, which however are usually reported in larger epicentral distances. The main innovation of this paper is emphasis to smaller epicentral distances. We studied properties of the wave using synthetic seismograms. The wave has a normal dispersion. Azimuthal and distance dependence of the wave partially explains its presence at 46 stations of 70 examined. Depth dependence shows that the studied earthquake was very efficient in the excitation of these waves just due to its shallow centroid depth (4.5 km).     

On the excitation of the earth's seismic normal modes

Summary The excitation of the earth's normal modes is formulated as an initial value problem. The static state of the earth, stressed from its hydrostatic reference situation, is considered as the initial state. The initial state is relaxed, at the time of the earthquake, by the removal of the forces maintaining the departure from hydrostatic equilibrium. Expressions are derived for the coefficients giving the relative excitation of the individual modes for the cases where these forces are compensating volume forces or compensating tractions on the faces of a dislocation. It is demonstrated that a point slip dislocation has a body force equivalent in the form of a double couple with a deviatoric moment tensor. However, for a source with volume change no moment tensor equivalent can be found. The volume change, apart from an elastic effect which can be represented by an isotropic moment tensor, has a direct gravitational effect on the excitation. This effect is due to a balanced force field consisting of a point force at the source and a continuous distribution of volume forces throughout the earth. The latter distribution, if not taken into account, may give rise to artificial phases in the frequency spectrum of the normal modes.     

On the completeness of inertial wave modes in rotating annular channels

An important unanswered mathematical question in the theory of rotating fluids has been the completeness of the inviscid eigenfunctions which are usually referred to as inertial waves or inertial modes. We provide for the first time a mathematical proof for the completeness of the inertial modes in a rotating annular channel by establishing the completeness relation, or Parseval’s equality, for any piecewise continuous, differentiable velocity of an incompressible fluid.     

On the solution of approximated systems when using reduced component modes

The work described in this paper relates to the computation of vibrational characteristics of a system which consists of components connected together elastically. The vibrational characteristics of each component are known and these are used via matrix manipulation to find the system characteristics. This particular paper confines itself to examining the matrix manipulation in detail and it is shown that the mass normalized modal matrix [Ψ] and connection matrix [C] can be contracted in size without affecting the overall solution. When the system is approximated by deleting rows and columns from the involved matrices, it is confirmed analytically that which rows are deleted from the mass normalized modal matrix [Ψ] has no relevance to the approximate solution for the system.     

On a general fluid dynamical theory of discrete unstable spiral modes in disk-shaped galaxies

Abstract

A general fluid dynamical theory of discrete unstable spiral modes in disk-shaped galaxies is described. This formulation of modes includes a radiation boundary condition and an exact numerical treatment of the Poisson equation. Thus, the modes are maintained by an outward transport of angular momentum, but they may be composed of both leading and trailing waves. A numerical scheme based on this formulation is described, and examples of modes obtained with this scheme are presented. These examples compare favorably with calculations based on the original asymptotic theory of Bertin, Lau, Lin and Mark. The implications of the present formulation of modes in galactic models support the hypothesis of a quasi-stationary spiral structure.     

Delineating ground water recharge from leaking irrigation canals using water chemistry and isotopes

Across the Great Plains irrigation canals are used to transport water to cropland. Many of these canals are unlined, and leakage from them has been the focus of an ongoing legal, economic, and philosophical debate as to whether this lost water should be considered waste or be viewed as a beneficial and reasonable use since it contributes to regional ground water recharge. While historically there has been much speculation about the impact of canal leakage on local ground water, actual data are scarce. This study was launched to investigate the impact of leakage from the Interstate Canal, in the western panhandle of Nebraska, on the hydrology and water quality of the local aquifer using water chemistry and environmental isotopes. Numerous monitoring wells were installed in and around a small wetland area adjacent to the canal, and ground water levels were monitored from June 1992 until January 1995. Using the water level data, the seepage loss from the canal was estimated. In addition, the canal, the monitoring wells, and several nearby stock and irrigation wells were sampled for inorganic and environmental isotope analysis to assess water quality changes, and to determine the extent of recharge resulting from canal leakage. The results of water level monitoring within study wells indicates a rise in local ground water levels occurs seasonally as a result of leakage during periods when the canal is filled. This rise redirects local ground water flow and provides water to nearby wetland ecosystems during the summer months. Chemical and isotopic results were used to delineate canal, surface, and ground water and indicate that leaking canal water recharges both the surface alluvial aquifer and upper portions of the underlying Brule Aquifer. The results of this study indicate that lining the Interstate Canal could lower ground water levels adjacent to the canal, and could adversely impact the local aquifer.     

Impact of repository depth on residence times for leaking radionuclides in land-based surface water

The multiple scales of landscape topography produce a wide distribution of groundwater circulation cells that control the hydro-geological environments surrounding geological repositories for nuclear waste. The largest circulation cells tend to discharge water into major river reaches, large freshwater systems or the nearby Baltic Sea. We investigated numerically the release of radionuclides from repositories placed in bedrock with depths between 100 to 2000 meters in a Swedish coastal area and found that leakage from the deeper positions emerges primarily in the major aquatic systems. In effect, radionuclides from the deeper repositories are more rapidly transported towards the Sea by the stream system compared to leakage from more shallow repositories. The release from the shallower repositories is significantly retained in the initial stage of the transport in the (superficial) landscape because the discharge occurs in or near low-order streams with high retention characteristics. This retention and residence time for radioactivity in the landscape control radiological doses to biota and can, thus, be expected to constitute an essential part of an associated risk evaluation.     

Study of fault slip modes

We present the data of the laboratory experiments on studying the regularities of gradual transition from the stick-slip behavior to aseismic creeping on the interblock boundary. The experiments show that small variations in the material composition in the principal slip zones of the faults may cause a significant change in the fraction of seismic energy radiated during the dynamic unloading of the adjacent segment of the rock mass. The experiments simulate interblock sliding regimes with the values of the scaled kinetic energy differing by a few orders of magnitude and relatively small distinctions in the strength of the contacts and in the amplitude of the released shear stresses. The results of the experiments show that the slip mode and the fraction of the deformation energy that goes into the seismic radiation are determined by the ratio of two parameters—the stiffness of the fault and the stiffness of the enclosing rock mass. An important implication of the study for solving the engineering tasks is that for bringing a stressed segment of a fault or a crack into a slip mode with low-intensity radiation of seismic energy, the anthropogenic impact should be aimed at diminishing the stiffness of the fault zone rather than at releasing the excessive stresses.     

Quasi-free-decay magnetic modes in planetary cores

A new category of hydromagnetic waves in a rotating conducting fluid within a spherical shell geometry is investigated. These quasi-free-decay magnetic modes are based on particular solutions of the induction equation where the magnetic diffusion plays the central role. These solutions, normally only decaying with time, become propagative owing to the combined action of the background magnetic field and the rotation. The amplitude and sign of their azimuthal phase drift strongly depend on morphology and magnitude of the background magnetic field. The validity domain of these quasi-free-decay (QFD)-modes is related to the Elsasser number and is written as Λ???1. It follows that these modes dissipate quickly before propagating out. This restriction falls when the above criterion is no longer fulfilled (Λ?～?1), the corresponding modes evolving towards distorted QFD-modes. A systematic study of these QFD-modes is made in the limit of small Elsasser number (Λ???1), for the different symmetries allowed. Application to the Earth's and other planetary cores is then examined for an Elsasser number up to Λ?≈?O(1), in relation to the geomagnetic secular variation and the frozen-flux approximation.     

The effect of accidental sulphuric acid leaking on metal distributions in estuarine sediment of Patos Lagoon.

In August of 1998 the tanker BAHAMAS belonging to the Chem Oil Company containing 12,000 t of concentrated sulphuric acid, had an accident on board, after which estuarine water entered one of the compartments of the tanker, resulting in a vigorous exothermic reaction. The reaction of acid with the metallic interior hull of the ship and the accompanying heat and H2 production resulted in an imminent risk of explosion. To avoid an explosion, given the fact that neutralization was not possible, some of the cargo was discharged into the surrounding water. Neutralization was done in January 1999, after the acid concentration in the tanker had decreased and the concentrations of Fe, Cr and Ni remained elevated. Metal concentrations in bottom sediments showed significant modifications. Leached mercury migrated and redeposited downstream, reaching approximately 76 times the background values. Such an anomaly has a well expressed barrier character. The mechanism for redeposition of Hg and other metals probably followed the pattern: Downstream as a result of dilution and mixing with seawater the pH of acid-water increases, favouring adsorption and/or precipitation of metals. The leading edge of a geochemical barrier, at positions 7-9 of sampling sites (Fig. 1), is confirmed by pH variations in the water. The reestablishment of normal pH occurred after a short time due to the high buffering capacity of seawater and large natural dilution process. The concentration of metals in estuarine water during and after the accident showed insignificant anomalies.     

CQC modal combination rule for high-frequency modes

The CQC rule for modal combination is extended to include the quasi-static contribution of truncated modes and the effects of input narrow-bandedness and cut-off frequency. A simple measure of the error in approximating a high-frequency modal response by its quasi-static contribution is derived. The extended rule is applicable to structures with high-frequency modes and to seismic inputs which may not be regarded as wide band. Numerical examples demonstrate the significance of input bandwidth and cut-off frequency on modal cross-correlation coefficients, and on the error resulting from truncation of high-freqeuncy modes.     

Stability and stable modes of coastal fronts

Abstract

The stability of a single layer, geostrophic front of zero potential vorticity bounded by a vertical coast (wall) is investigated by means of a Rayleigh integral. It is proved that the flow of the density-driven current is stable at all wavenumbers provided the mean velocity of basic flow exceeds fL (where f is the Coriolis parameter and L is the distance between the wall and the free streamline). The frequency of the stable long waves is either zero or super-inertial.     

关于弹性应力波的三种传播模式

分析了小变形情况下变形体的平动变形和转动变形,利用连续介质的动量和动量矩守恒方程,应用各向同性线弹性体的本构关系和偶应力本构关系,提出无限弹性介质中存在体积波(压力波)、旋转波(偶应力波)和偏斜波(偏斜应力波).旋转波和偏斜波均满足形式一致的四阶波动方程,但引起不同的运动学行为和应力状态,四阶旋转波和偏斜波的传播速度不再是依赖材料参数的常值.不计旋转变形时,各向同性弹性固体中只有体积波和二阶偏斜波的传播,这时传播速度均为常数.在3种波动模式中,体积波与传统弹性应力波理论完全一致,四阶偏斜波可退化为二阶偏斜波,但后者不同于传统应力波理论中以位移作为波函数的二阶剪切波.从变形运动学与内力的关系以及能量传播的角度分析了传统应力波理论中关于二阶旋转波和二阶剪切波存在的问题.     

Normal modes of black hole accretion disks

This article studies the hydrodynamical problem of normal modes of small adiabatic oscillations of relativistic barotropic thin accretion disks around black holes (and compact weakly magnetic neutron stars). Employing Wentzel-Kramers-Brillouin (WKB) techniques, we obtain the eigenfrequencies and eigenfunctions of the modes for different values of the mass and angular momentum of the central black hole. We discuss the properties of the various types of modes and show for the first time that modes covering the whole extension of the disk (full-disk p-modes) can exist within the studied thin disk model. However, these modes have relatively short wavelengths.     

Three distinct reversing modes in the geodynamo

The data that describe the long-term reversing behavior of the geodynamo show strong and sudden changes in magnetic reversal frequency. This concerns both the onset and the end of superchrons and most probably the occurrence of episodes characterized by extreme geomagnetic reversal frequency (>10–15 rev./Myr). To account for the complexity observed in geomagnetic reversal frequency evolution, we propose a simple scenario in which the geodynamo operates in three distinct reversing modes: i—a “normal” reversing mode generating geomagnetic polarity reversals according to a stationary random process, with on average a reversal rate of ～3 rev./Myr; ii—a non-reversing “superchron” mode characterizing long time intervals without reversal; iii—a hyper-active reversing mode characterized by an extreme geomagnetic reversal frequency. The transitions between the different reversing modes would be sudden, i.e., on the Myr time scale. Following previous studies, we suggest that in the past, the occurrence of these transitions has been modulated by thermal conditions at the core-mantle boundary governed by mantle dynamics. It might also be possible that they were more frequent during the Precambrian, before the nucleation of the inner core, because of a stronger influence on geodynamo activity of the thermal conditions at the core-mantle boundary.     

Coupling between waveguide modes and field line resonances

WKBJ methods are often used to study the excitation of waveguide modes and field line resonances in the magnetosphere. In this paper, these methods are used to study the nature of the reflection and transmission process at a turning point when conditions are favourable for strong excitation of the field line resonance. The accuracy and region of validity of such WKBJ solutions are discussed. In regions where they are not valid, a series solution of the relevant equations is given. The wave fields can be found to great accuracy by matching WKBJ solutions across the turning point and resonance using this series solution. This allows the calculation of the Stokes constant which does not show Airy function behaviour as often assumed. The Stokes constant is effectively a reflection coefficient referred to the turning point. Its phase, when the transmission to the resonance is strong, is nearer to 180° than 90° as often assumed. Its magnitude allows the fraction of incident energy which is transmitted to the resonance to be calculated. These results are expected to be useful in studying the behaviour of waveguide modes.     

不同加载方式下黄土累积变形特征研究

土体动力特性主要与其受力状态、土体加载类型、试验控制方式以及荷载等参数有关。以往非饱和黄土累积变形试验研究通过单向动应力加载方式开展相关研究,并不能真实反映土体受力状态。现运用WF-12440型动三轴-空心圆柱扭剪试验系统,通过单向荷载和双向荷载两种不同的动应力加载方式,进行黄土的长期重复载荷试验,对比不同动应力加载方式对非饱和黄土累积变形发展特征的影响。试验结果发现：黄土在单向荷载或双向荷载作用下,其软化指数均随着循环次数的增加,呈非线性减小的趋势,在相同循环振次时,土体软化指数随着动应力幅值的增大而减小;黄土在不同加载方式作用下,软化指数减小速率在循环振次100次以内较快,后期随振次增加,土体软化速率逐渐趋于平稳;黄土在双向荷载作用下,当轴向动荷载较小时,径向荷载的施加,将加速黄土的软化程度,随着轴向动荷载的增大,径向荷载对黄土软化的影响逐渐减弱,其对于黄土变形的影响存在一临界轴向循环动应力。本研究的开展,可更准确地进行不同荷载耦合变化下,黄土场地震陷评估或路基土体变形计算,实现有效控制路基的整体稳定和工后沉降量,以期根据路基土参数预测后期沉降,极具重要的科研和工程意义。     

Focusing phenomenon and near-trapped modes of SH waves

In this study, the null-field boundary integral equation method (BIEM) and the image method are used to solve the SH wave scattering problem containing semi-circular canyons and circular tunnels. To fully utilize the analytical property of circular geometry, the polar coordinates are used to expand the closed-form fundamental solution to the degenerate kernel, and the Fourier series is also introduced to represent the boundary density. By collocating boundary points to match boundary condition on the boundary, a linear algebraic system is constructed. The unknown coefficients in the algebraic system can be easily determined. In this way, a semi-analytical approach is developed. Following the experience of near-trapped modes in water wave problems of the full plane, the focusing phenomenon and near-trapped modes for the SH wave problem of the half-plane are solved, since the two problems obey the same mathematical model. In this study, it is found that the SH wave problem containing two semi-circular canyons and a circular tunnel has the near-trapped mode and the focusing phenomenon for a special incident angle and wavenumber. In this situation, the amplification factor for the amplitude of displacement is over 300.