The α-effect in 2D fast dynamos

We look at the large-scale dynamo properties of spatially periodic, time dependent, helical 2D flows of the form u(x, t)?=?(? _y ?ψ?(x, y, t), ?? _x ?ψ?(x, y, t), ?ψ (x, y, t). These flows act as kinematic fast dynamos and are able to generate a mean magnetic field uniform and constant in the xy-plane but whose direction varies periodically along z with wavenumber k. Using Mean Field Electrodynamics, the generation mechanism can be understood in terms of a k-dependent α-effect, which depends on the magnetic Reynolds number, R _m . We calculate this effect for different motions and investigate how its limit as k?→?0 depends on R _m and on the properties of the flows such as their spatial structure or correlation time. This work generalises earlier studies based on 2D steady flows to motions with time dependence.     

The Displacement and Strain Field of Three-dimensional Rheologic Model of Earthquake Preparation

Based on the three-dimensional elastic inclusion model proposed by Dobrovolskii, we developed a rheological inclusion model to study earthquake preparation processes. By using the Corresponding Principle in the theory of rheologic mechanics, we derived the analytic expressions of viscoelastic displacement U(r, t) , V(r, t) and W(r, t), normal strains ε_xx (r, t), ε_yy (r, t) and ε_zz (r, t) and the bulk strain θ (r, t) at an arbitrary point (x, y, z) in three directions of X axis, Y axis and Z axis produced by a three-dimensional inclusion in the semi-infinite rheologic medium defined by the standard linear rheologic model. Subsequent to the spatial-temporal variation of bulk strain being computed on the ground produced by such a spherical rheologic inclusion, interesting results are obtained, suggesting that the bulk strain produced by a hard inclusion change with time according to three stages (α, β, γ) with different characteristics, similar to that of geodetic deformation observations, but different with the results of a soft inclusion. These theoretical results can be used to explain the characteristics of spatial-temporal evolution, patterns, quadrant-distribution of earthquake precursors, the changeability, spontaneity and complexity of short-term and imminent-term precursors. It offers a theoretical base to build physical models for earthquake precursors and to predict the earthquakes.     

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

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

Numerical simulations of transport of non-ergodic solute plumes in heterogeneous aquifers

Transport of non-ergodic solute plumes by steady-state groundwater flow with a uniform mean velocity, μ, were simulated with Monte Carlo approach in a two-dimensional heterogeneous and statistically isotropic aquifer whose transmissivity, T, is log-normally distributed with an exponential covariance. The ensemble averages of the second spatial moments of the plume about its center of mass, <S _{i i} (t)>, and the plume centroid covariance, R _{i i} (t) (i=1,2), were simulated for the variance of Y=log T, σ _Y ²=0.1, 0.5 and 1.0 and line sources normal or parallel to μ of three dimensionless lengths, 1, 5, and 10. For σ _Y ²=0.1, all simulated <S _{i i} (t)>−S _{i i} (0) and R _{i i} (t) agree well with the first-order theoretical values, where S _{i i} (0) are the initial values of S _{i i} (t). For σ _Y ²=0.5 and 1.0 and the line sources normal to μ, the simulated longitudinal moments, <S ₁₁(t)>−S ₁₁(0) and R ₁₁(t), agree well with the first-order theoretical results but the simulated transverse moments <S ₂₂(t)>−S ₂₂(0) and R ₂₂(t) are significantly larger than the first-order values. For the same two larger values of σ _Y ² but the line sources parallel to μ, the simulated <S ₁₁(t)>−S ₁₁(0) are larger than but the simulated R ₁₁ are smaller than the first-order values, and both simulated <S ₂₂(t)>−S ₂₂(0) and R ₂₂(t) stay larger than the first-order values. For a fixed value of σ _Y ², the summations of <S _{i i} (t)>−S _{i i} (0) and R _{i i} , i.e., X _{i i} (i=1,2), remain almost the same no matter what kind of source simulated. The simulated X ₁₁ are in good agreement with the first-order theory but the simulated X ₂₂ are significantly larger than the first-order values. The simulated X ₂₂, however, are in excellent agreement with a previous modeling result and both of them are very close to the values derived using Corrsin's conjecture. It is found that the transverse moments may be significantly underestimated if less accurate hydraulic head solutions are used and that the decreasing of <S ₂₂(t)>−S ₂₂(0) with time or a negative effective dispersivity, defined as , may happen in the case of a line source parallel to μ where σ _Y ² is small.     

Numerical simulations of transport of non-ergodic solute plumes in heterogeneous aquifers

Transport of non-ergodic solute plumes by steady-state groundwater flow with a uniform mean velocity, μ, were simulated with Monte Carlo approach in a two-dimensional heterogeneous and statistically isotropic aquifer whose transmissivity, T, is log-normally distributed with an exponential covariance. The ensemble averages of the second spatial moments of the plume about its center of mass, <S _{i i} (t)>, and the plume centroid covariance, R _{i i} (t) (i=1,2), were simulated for the variance of Y=log T, σ _Y ²=0.1, 0.5 and 1.0 and line sources normal or parallel to μ of three dimensionless lengths, 1, 5, and 10. For σ _Y ²=0.1, all simulated <S _{i i} (t)>−S _{i i} (0) and R _{i i} (t) agree well with the first-order theoretical values, where S _{i i} (0) are the initial values of S _{i i} (t). For σ _Y ²=0.5 and 1.0 and the line sources normal to μ, the simulated longitudinal moments, <S ₁₁(t)>−S ₁₁(0) and R ₁₁(t), agree well with the first-order theoretical results but the simulated transverse moments <S ₂₂(t)>−S ₂₂(0) and R ₂₂(t) are significantly larger than the first-order values. For the same two larger values of σ _Y ² but the line sources parallel to μ, the simulated <S ₁₁(t)>−S ₁₁(0) are larger than but the simulated R ₁₁ are smaller than the first-order values, and both simulated <S ₂₂(t)>−S ₂₂(0) and R ₂₂(t) stay larger than the first-order values. For a fixed value of σ _Y ², the summations of <S _{i i} (t)>−S _{i i} (0) and R _{i i} , i.e., X _{i i} (i=1,2), remain almost the same no matter what kind of source simulated. The simulated X ₁₁ are in good agreement with the first-order theory but the simulated X ₂₂ are significantly larger than the first-order values. The simulated X ₂₂, however, are in excellent agreement with a previous modeling result and both of them are very close to the values derived using Corrsin's conjecture. It is found that the transverse moments may be significantly underestimated if less accurate hydraulic head solutions are used and that the decreasing of <S ₂₂(t)>−S ₂₂(0) with time or a negative effective dispersivity, defined as , may happen in the case of a line source parallel to μ where σ _Y ² is small.     

Curie Point Depths of Albania Inferred from Ground Total Field Magnetic Data

Ground total magnetic field data of Albania were used to produce estimates of the Curie point isotherm. The strategy followed was to estimate the depth to the bottom of the deepest magnetic sources. Firstly, the average depth to the top of the deepest crustal block, z_t, was computed by linear fitting to the second lowest-frequency segment of the azimuthally averaged power spectrum of the total magnetic field data. Then, the depth to the centroid of the deepest crustal block, z₀, was computed by linear fitting to the lowest-frequency segment of the azimuthally averaged power spectrum of a distribution of magnetic dipoles. Finally, the depth to the bottom, the inferred Curie point depth, z_b, was calculated from z_b=2z₀−z_t. Curie depth estimates for Albania vary from about 17 to 25.5 km (below sea level). These results are consistent with the depths inferred by extrapolating geothermal gradient and heat-flow values, suggesting that the Curie point depth analysis is useful to estimate the regional thermal structure. It also suggests that the approach was valid and that ground total magnetic field data can be used for this purpose.     

Generation of superDreicer electric fields in the solar chromosphere

The electric field generation at the front of the current pulse, which originates in a coronal magnetic loop owing to the development of the Rayleigh–Taylor magnetic instability at loop footpoints, has been considered. During the τ_A ≈ l/V _A ≈ 5?25 s time (where l is the plasma plume height entering a magnetic loop as a result of the Rayleigh–Taylor instability), a disturbance related to the magnetic field tension B _?(r,t), “escapes” the instability region with the Alfvén velocity in this case. As a result, an electric current pulse I_z(z ? V _A t), at the front of which an induction magnetic field E _z, which is directed along the magnetic tube axis and can therefore accelerate particles, starts propagating along a magnetic loop with a characteristic scale of Δξ ≈ l. In the case of sufficiently large currents, when B _? ²/8π > p, an electric current pulse propagates nonlinearly, and a relatively large longitudinal electric field originates E _z ≈ 2I _z ³ V _A/c ⁴a²B_z ²l, which can be larger than the Dreicer field, depending on the electric current value.     

On the effects of a bumpy core-mantle interface

Motivated by the high degree of correlation between the variable parts of the magnetic and gravitational potentials of the Earth discovered by Hide and Malin (using a harmonic analysis approach and utilizing the geomagnetic data) when one field is suitably displaced relative to the other, Moffatt and Dillon (1976) studied a simple planar model in an attempt to find a quantitative explanation for the suggestion that this high degree of correlation may be due to the influences produced by bumps on the core-mantle interface. Moffatt and Dillon assumed that the core-mantle interface was z = η(x) where |?η/?χ| ? 1 and such that in the core [z < η(x)] a uniform flow (U₀, 0, 0) prevails in the presence of a uniform ‘toroidal’ field (B₀, 0, 0); (here z is the vertical coordinate and x is the eastward distance). The whole system rotates uniformly about the vertical with angular velocity Ω. The present work extends the model discussed by Moffatt and Dillon to include a horizontal component of angular velocity Ω_H and a uniform small poloidal field B_p. In addition, the uniform toroidal field is here replaced by one which vanishes everywhere in the mantle and increases linearly, from zero on the interface, with z. It is shown that the presence of Ω_H and B_p, together with the present choice of toroidal magnetic field, has a profound effect both on the correlation between the variable parts of the magnetic and gravitational fields of the Earth, and on how far the disturbances caused by the topography of the interface [which is necessarily three-dimensional i.e. z = η(x, y) here] can penetrate into the liquid core. In particular it is found that the highest value of the correlation function is +0.79 which corresponds to a situation in which the magnetic potential is displaced both latitudinally and longitudinally relative to the gravitational potential.     

Enhanced velocity estimation using gridded tomography in complex chalk

The theme of the 2003 EAGE/SEG imaging workshop concerned the contrast between different philosophies of ‘model building’: whether an explicit, user‐determined model should be imposed throughout the processing, with user updates at each step; or alternatively, whether user intervention should be kept to a minimum so as to avoid preconceived bias, and instead to allow the data itself to guide some heuristic process to converge to an optimal solution. Here we consider a North Sea study where our initial approach was to build the subsurface model using interpreted horizons as a guide to the velocity update. This is common practice in the North Sea, where the geology ‘lends itself’ to a layer‐based model representation. In other words, we encourage preconceived bias, as we consider it to be a meaningful geological constraint on the solution. However, in this instance we had a thick chalk sequence, wherein the vertical compaction gradient changed subtly, in a way not readily discernible from the seismic reflection data. As a consequence, imposing the explicit top and bottom chalk horizons, with an intervening vertical compaction gradient (of the form v(x, y, z) =v₀(x, y) +k(x, y).z), led to a misrepresentation of the subsurface. To address this issue, a gridded model building approach was also tried. This relied on dense continuous automatic picking of residual moveout in common‐reflection point gathers at each iteration of the model update, followed by gridded tomography, resulting in a smoothly varying velocity field which was able to reveal the underlying local changes within the chalk.     

APPROXIMATION OF THE GRAVITATIONAL ATTRACTION OF GEOLOGICAL BODIES*

The gravitational attraction produced by a geological body of irregular shape can be easily determined by dividing it into cubes of small size. The exact expression of the effect of a cube is very complicated, but it can be calculated by using an electronic computer. 4851 values of the gravitational attraction were determined for different positions of a cube with the side of 2l and the center in M(x₀, y₀, z₀), for x₀∈[0;20], y₀∈[0;20] and z₀∈[0;10]. Using these values, templates in double logarithmic representation were drawn, with x₀ and z₀ as parameters and y₀ as abscissa, and with x₀ and y₀ as parameters and z₀ in abscissa; this double set of templates permits a good interpolation for all cubes in the considered domain of M. The use of templates was tested to approximate the effect produced by a theoretical model of spherical shape and in a real case of a three-dimensional salt body of known shape based on a large number of boreholes. In both cases very good results were obtained.     

On the topographic coupling at the core-mantle interface

The mean tangential stresses at a corrugated interface between a solid, electrically insulating mantle and a liquid core of magnetic diffusivity λ are calculated for uniform rotation of both mantle and core at an angular velocity Ω in the presence of a corotating magnetic field B. The core and mantle are assumed to extend indefinitely in the horizontal plane. The interface has the form z = η(x, y), where z is the upward vertical distance and x, y are the zonal and latitudinal distances respectively. The function η(x, y) has a planetary horizontal length scale (i.e. of the order of the radius of the Earth) and small amplitude and vertical gradient. The liquid core flows with uniform mean zonal velocity U₀ relative to the mantle. Ω and B possess vertical and horizontal components.The vertical (poloidal) component B_p is uniform and has a value of 5 G while the horizontal (toroidal) field B_T = B_pαz, where α is a constant. When |α| ? 1, the mean horizontal stresses are found to have the same order of magnitude (10^?2 N m^?2) as those inferred from variations in the decade fluctuations in the length of the day, although the exact numerical values depend on the orientation of Ω as well as on the wavenumbers in the zonal and latitudinal directions.The influence of the steepness (as measured by α) of the toroidal field on the stresses is investigated to examine whether the constraint that the mean horizontal stresses at the core-mantle interface be of the order of 10^?2 N m^?2 might provide a selection mechanism for the behaviour of the toroidal field in the upper reaches of the outer core of the Earth. The results indicate that the restriction imposed on α is related to the value assigned to the toroidal field deep into the core. For example, if |α| ? 1 then the tangential stresses are of the right order of magnitude only if the toroidal field is comparable with the poloidal field deep in the core.     

A multivariate gamma distribution arising from a Markov model

A Markov chain{X _t }, which has been useful for modelling in hydrology, can be specified by the Laplace transform (LT) of the conditional p.d.f. ofX _t+1 givenX _t =x _t , which is assumed to be of the exponential formH()exp{-G()x _t }. For appropriate choice ofH andG the marginal distribution ofX _t is the (univariate) gamma distribution. In this case, the joint p.d.f. ofX _t +1,...,X _t+n and its LT, are obtained, and this is extended to a seasonal version of the chain. A simple method of generating observations from these multivariate gamma distributions is noted, and the joint LT is applied to the problem of determining moments of weighted sums of such variables.     

Alignment silkworms as seismic animal anomalous behavior (SAAB) and electromagnetic model of a fault: a theory and laboratory experiment

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THE INDUCTIVE RESPONSE OF A HORIZONTAL CONDUCTING CYLINDER BURIED IN A UNIFORM EARTH FOR A UNIFORM INDUCING FIELD*

The inductive response of a conducting horizontal cylinder embedded in a uniform earth is studied using numerical results obtained for an analytical solution for the problem of a conducting cylinder buried in a homogeneous earth for the case of a uniform inducing field. A check of the validity of the numerical results is made by a comparison with analogue model measurements for a number of cases. Numerical results for a range of cylinder radii (a = 1–10 km), depths of burial (d= 0–4 km), conductivity contrasts (σ₂= 10^?2-10 Sm^?1), and source frequencies (f= 10^?1-10^?4 Hz) of interest in the interpretation of magnetotelluric field measurements are presented. The results indicate that for a uniform inducing field the conductivity and depth of burial of a horizontal cylindrical inhomogeneity are best determined through a measurement of the amplitudes H_y, H_z and E_x and the phases φ_y and Ψ_x.     

The correlation between gravitational and geomagnetic fields caused by interaction of the core fluid motion with a bumpy core-mantle interface

The correlation discovered by Hide and Malin between the variable parts of the Earth's gravitational field and magnetic field (suitably displaced in longitude) was tentatively and qualitatively explained by them in terms of the influence on both fields of irregularities (or “surface bumps”) at the core-mantle interface. In this paper, a quantitative analysis of this phenomenon is developed, through study of an idealised problem in which conducting fluid occupying the region z < η(x) flows over the surface z = η(x) in the presence of a magnetic field (B₀,0,0), the whole system rotating with angular velocity (0,0,Ω). It is assumed that $\text{|η′(x)| « 1}$ so that perturbation methods are applicable. Determination of the magnetic potential in the “mantle” region z < η(x) requires solution of the full hydromagnetic problem in the fluid. It is shown that three wave modes are excited, two of which (for values of the parameters of the problem of geophysical interest) have a boundary layer character. Phase interactions between these modes lead to a shift and a distortion of the magnetic pattern relative to the gravitational pattern. The correlation between the gravitational potential and the magnetic potential (shifted by a distance x₀) is determined on the plane $\text{z = d (d}\text{a}\text{?}\text{|η|)}$ as a function of x₀/d and the curves obtained are qualitatively similar to that based on the observed data; the maximum correlation obtained varies between 0.67 and 1, depending on values of the parameters of the problem, and is about 0.72 for reasonable estimates of these parameters in the geophysical context.     

Yield and overflow from a finite stochastic reservoir

The paper discusses the overflow (spillage) and yield rates and the total overflow and total yield over a specified time from a finite discrete stochastic reservoir, in which the yieldY _t during the working interval (t,t+1) is a function of the storageZ _t at timet, the inflow sequence {X _t } being IID.The distribution vector of the spillage rate at timet is a telescoped version of the distribution of a certain Markovian variable whose transition matrix is derived. Formulae are given for the distribution of the total spillageW _h given suitable initial conditions, forh=1,2,3; and a simple expression derived forE(W _h ).The distribution of the yield rateY _t is a trivial modification of the storage distribution. As for the total yieldR _t =Y ₁+...+Y _t , it is shown that the bivariate sequence {R _t ,Z _t } is first-order Markovian, whereZ _t is the storage at timet. The transition matrix of this process is obtained and the method of evaluating the marginal distribution of the total yieldR _t is exemplified.     

A statistical study of the interplanetary magnetic field control of sporadic E-layer occurrence in the southern polar cap ionosphere

The influence of the interplanetary magnetic field (IMF) on the occurrence of sporadic E (Es)-layers in the southern polar cap ionosphere has been investigated. We statistically analysed ionogram and Doppler velocity observations made using a HF digital ionosonde located at Casey, Antarctica (66.3°S, 110.5°E; 81°S magnetic latitude) during the two summer campaign intervals 1 January to 18 February, and 1 November to 31 December 1997. The ionogram and Doppler velocity measurements were used to determine the Es-occurrence and electric field vectors (assuming E×B/B² drift), respectively. Concurrent IMF data were obtained from measurements made on board the Wind spacecraft. First, the gross properties of the IMF dependence of Es-formation were obtained: the occurrence rate was higher for negative B_y and/or positive B_z, and lower for positive B_y and/or negative B_z. To reconcile these gross properties with the electric field theory of Es-layer formation, the detailed diurnal variation of both Es-occurrence and the ionospheric electric field were obtained for different orientations of the IMF. The main statistical results are that: (1) the B_y component mainly controls the occurrence of the midnight Es-layers through its influence on the corresponding South West electric field; and (2) the B_z component mainly controls the occurrence of the evening Es-layers. However, the change in the occurrence rate for evening Es-layers was not related to the strength of the associated North West and North East electric fields. The total occurrence of Es-layers depended more on B_y than on B_z, owing to the dominance of B_y-controlled midnight Es-layers in the occurrence distribution. Nevertheless, the dependence of Es-occurrence on B_z was important. We suggest that the increase in Es-occurrence for positive B_z might be explained by the intermittent production of lower F-region ionisation by polar showers and squalls, which also increase in frequency and intensity for positive B_z. The importance of metallic ion transport within the ionosphere is also considered.     

Yield and overflow from a finite stochastic reservoir

The paper discusses the overflow (spillage) and yield rates and the total overflow and total yield over a specified time from a finite discrete stochastic reservoir, in which the yieldY _t during the working interval (t,t+1) is a function of the storageZ _t at timet, the inflow sequence {X _t } being IID.The distribution vector of the spillage rate at timet is a telescoped version of the distribution of a certain Markovian variable whose transition matrix is derived. Formulae are given for the distribution of the total spillageW _h given suitable initial conditions, forh=1,2,3; and a simple expression derived forE(W _h ).