Second-order Discontinuities Research Articles

Second-Order Discontinuities for Simple and Double Layers

Simple and double layers first appeared in electrostatics and later found various applications in mathematical physics. In this paper, we present the jump discontinuity conditions for their second-order spatial derivatives.

Neuquén Group (Upper Cretaceous): A case of underfilled-overfilled cycles in an Andean foreland basin, Neuquén basin, Argentina

The Nenquén Group was deposited during a period dominated by the Cretaceous Greenhouse and can be divided in three cycles correlated with large-scale changes in the evolution of the Andean foreland basin. The filling of the Neuquén Group is constituted by a complete cycle and two incomplete cycles of underfilled-overfilled, separated by first-order discontinuities assigned to the uplift of the Agrio fold-and-thrust belt during the Chasca/Catequil, Mid Ocean Ridge (CCMOR) collision, coinciding with first-order climatic changes within the Cretaceous greenhouse cycle.The Candeleros Formation in the base of this group was deposited in late underfilled conditions, showing prominent forebulge zones. It is demonstrated that during the Albian, with the cratonward migration of the uplifting forebulge zones, the axis of backbulge zones also migrated cratonwards and a wide uplifted forebulge zone was formed. On top, the Huincul Formation was deposited in an overfilled period without orogenic load, while the Cerro Lisandro Formation was deposited in early underfilled conditions with orogenic load. The Río Neuquén Subgroup started with a late underfilled period (Portezuelo Formation -second-order discontinuity), and after wards the Plottier Formation was deposited in an overfilled period without orogenic load. Finally, the Río Colorado Subgroup was deposited under late and early underfilled conditions (Bajo de la Carpa and Anacleto Formations respectively).

Dynamical transition in the temporal relaxation of stochastic processes under resetting.

A stochastic process, when subject to resetting to its initial condition at a constant rate, generically reaches a nonequilibrium steady state. We study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems. We show that as time progresses an inner core region around the resetting point reaches the steady state, while the region outside the core is still transient. The boundaries of the core region grow with time as power laws at late times with new exponents. Alternatively, at a fixed spatial point, the system undergoes a dynamical transition from the transient to the steady state at a characteristic space-dependent timescale t(*)(x). We calculate analytically in several examples the large deviation function associated with this spatiotemporal fluctuation and show that, generically, it has a second-order discontinuity at a pair of critical points characterizing the edges of the inner core. These singularities act as separatrices between typical and atypical trajectories. Our results are verified in the numerical simulations of several models, such as simple diffusion and fluctuating one-dimensional interfaces.

Carlsberg Ridge and Mid-Atlantic Ridge: Comparison of slow spreading centre analogues

Eighty per cent of all mid-ocean spreading centres are slow. Using a mixture of global bathymetry data and ship-board multibeam echosounder data, we explore the morphology of global mid-ocean ridges and compare two slow spreading analogues: the Carlsberg Ridge in the north-west Indian Ocean between 57°E and 60°E, and the Kane to Atlantis super-segment of the Mid-Atlantic Ridge between 21°N and 31°N. At a global scale, mid-ocean spreading centres show an inverse correlation between segment length and spreading rate with segmentation frequency. Within this context, both the Mid-Atlantic Ridge super-segment and Carlsberg Ridge are similar: spreading at 22 and 26mm/yr full rates respectively, being devoid of major transform faults, and being segmented by dextral, non-transform, second-order discontinuities. For these and other slow spreading ridges, we show that segmentation frequency varies inversely with flank height and ridge axis depth. Segments on both the Mid-Atlantic Ridge super-segment and Carlsberg Ridge range in aspect ratio (ridge flank height/axis width), depth and symmetry. Segments with high aspect ratios and deeper axial floors often have asymmetric rift flanks and are associated with indicators of lower degrees of melt flux. Segments with low aspect ratios have shallower axial floors, symmetric rift flanks, and evidence of robust melt supply. The relationship between segmentation, spreading rate, ridge depth and morphology, at both a global and local scale, is evidence that rates of melting of the underlying mantle and melt delivery to the crust play a significant role in determining the structure and morphology of slow spreading mid-ocean ridges.

Second-order contour discontinuities in segmentation and shape representation

Second-order contour discontinuities in segmentation and shape representation

The dynamic contact of an impactor and an elastic orthotropic plate when there are propagating thermoelastic waves

The onset and propagation of longitudinal and transverse elastic waves and a temperature wave in a circular plate possessing cylindrical anisotropy is investigated. As a result of this the plate becomes deformed when there is a temperature shock or shock interaction with the heated body. The plate is assumed to be fairly extended and reflected waves are ignored when calculating the stress state. The onset of a temperature wave is possible if the non-classical hyperbolic-type heat-conduction equation is employed. To determine the displacements of points outside the contact region a ray method is employed based on the use of power series with respect to time and a spatial coordinate as well as compatibility conditions. The displacements and internal forces are determined in the form of sections of ray series apart from integration constants, which are found from the contact conditions. The temperature effect on the propagating wave is investigated in the form of second-order discontinuity surfaces.

Zona de fratura de Vitória-Trindade no Oceano Atlântico sudeste e suas implicações tectônicas

A integração de dados geológicos e geofísicos permitiu um melhor entendimento da inter-relação entre estruturas oceânicas e continentais. Fenômenos tectônicos e magmáticos do Meso-Cenozóico do Sudeste Brasileiro podem ter sido condicionados pelo prolongamento das zonas de fratura oceânicas em direção à margem continental. O foco deste trabalho é a evolução da Zona de Fratura de Vitória-Trindade (ZFVT). A Zona de Fratura de Vitória-Trindade (ZFVT), localizada na latitude de 18º40' na Cordilheira Mesoatlântica (CMA), foi mapeada utilizando-se sísmica de reflexão, gravimetria e isócronas magnéticas. A ZFVT na CMA corresponde a um deslocamento axial em torno de 6 km, configurando uma descontinuidade de segunda ordem. O segmento assísmico da ZFVT é definido por uma depressão gravimétrica de direção E-W. Sua extensão para oeste corresponde à Cadeia Vitória-Trindade, representada por altos topográficos individualizados, que limitam desníveis crustais de mais de 400 m e províncias sedimentares de características e espessuras distintas. Esta região cortada pela intrusão de diques (alguns aflorando na superfície) e sills, indica instabilidade tectônica com a movimentação de falhas e ocorrência de estruturas transpressivas (estruturas em flor), elevando toda a coluna sedimentar incluindo o assoalho oceânico. Estas observações sugerem reativações recentes da crosta oceânica com esforços alternantes, ora compressivos ora distensivos, associados a vulcanismos e falhamentos normais. Evidências geológicas e geofísicas, além do mapeamento do alinhamento de epicentros de terremotos, sugerem que o prolongamento da ZFVT está relacionado ao Alto de Vitória e ao tectonismo e magmatismo na porção emersa, representado pelo Lineamento Sismo-Magmático Alcalino Trindade-Paxoréu / Alto do Paranaíba. Propõe-se que a ZFVT atuou e continua atuando como conduto para o magmatismo da Pluma Mantélica de Trindade.

The influence of ridge migration on the magmatic segmentation of mid-ocean ridges.

The Earth's mid-ocean ridges display systematic changes in depth and shape, which subdivide the ridges into discrete spreading segments bounded by transform faults and smaller non-transform offsets of the axis. These morphological changes have been attributed to spatial variations in the supply of magma from the mantle, although the origin of the variations is poorly understood. Here we show that magmatic segmentation of ridges with fast and intermediate spreading rates is directly related to the migration velocity of the spreading axis over the mantle. For over 9,500 km of mid-ocean ridge examined, leading ridge segments in the 'hotspot' reference frame coincide with the shallow magmatically robust segments across 86 per cent of all transform faults and 73 per cent of all second-order discontinuities. We attribute this relationship to asymmetric mantle upwelling and melt production due to ridge migration, with focusing of melt towards ridge segments across discontinuities. The model is consistent with variations in crustal structure across discontinuities of the East Pacific Rise, and may explain variations in depth of melting and the distribution of enriched lavas.

Open fissure mineralization at 2600 m depth in Long Valley Exploratory Well (California) – insight into the history of the hydrothermal system

Long Valley Exploratory Well, drilled into the Resurgent Dome at Long Valley Caldera (California) to explore the potential of geothermal power in an active magmatic system, achieved temperatures of only ca. 100°C at 2500–3000 m depth, well below the range expected atop an active magma chamber. Open fissures encountered at 2600 m depth are coated by mm-sized idiomorphic quartz crystals with first- and second-order growth discontinuities. Specific growth defects indicating rapid crystallization reflect sudden changes in SiO 2 supersaturation. Fluid inclusions contain low salinity (0–5 wt% NaCl) and low CO 2 (<3 mole%) aqueous fluids, with V–L homogenization temperatures of 300–350°C, indicating trapping at more than 200°C above the ambient temperatures measured within the borehole today. Fluid composition and inclusion density varies between and within the growth zones, reflecting progressive changes in the hydrothermal system during crystallization. Episodic crystallization from supersaturated fluids is interpreted to reflect sudden changes in the convection pattern, presumably induced by seismic activity, with a more recent and dramatic reorganization resulting in convective cooling. The quartz crystals are sensitive recorders of the earlier higher temperature history, unaffected by the present-day situation.

Spectral-element simulations of global seismic wave propagation-I. Validation

SUMMARY We use a spectral-element method to simulate seismic wave propagation throughout the entire globe. The method is based upon a weak formulation of the equations of motion and combines the flexibility of a finite-element method with the accuracy of a global pseudospectral method. The finite-element mesh honours all first- and second-order discontinuities in the earth model. To maintain a relatively constant resolution throughout the model in terms of the number of grid points per wavelength, the size of the elements is increased with depth in a conforming fashion, thus retaining a diagonal mass matrix. In the Earth’s mantle and inner core we solve the wave equation in terms of displacement, whereas in the liquid outer core we use a formulation based upon a scalar potential. The three domains are matched at the inner core and core‐ mantle boundaries, honouring the continuity of traction and the normal component of velocity. The effects of attenuation and anisotropy are fully incorporated. The method is implemented on a parallel computer using a message passing technique. We benchmark spectral-element synthetic seismograms against normal-mode synthetics for a spherically symmetric reference model. The two methods are in excellent agreement for all body- and surface-wave arrivals with periods greater than about 20 s.

Experiments in Fingertip Perception of Surface Discontinuities

In this paper, we present the results of experiments that explore the ability of the human fingertip to detect haptically simulated first-order and second-order surface discontinuities. A single finger, planar motion fingertip haptic display (FHD) developed at the University of Washington was used by multiple test operators to kinesthetically trace haptically simulated surfaces under a variety of treatment conditions. Experiment variables included the magnitude and type of discontinuity, as well as the stiffness and damping terms used in the haptic simulation algorithm. Test operators were asked to haptically locate a discontinuity on a simulated surface for each treatment condition with the location accuracy being used as the experiment performance metric. The results reveal how the surface discontinuity detection ability is degraded by control gains that are either too low or too high, indicating that a given haptic simulation system may have an optimum set of control gains that will yield the best performance for this type of task.

Near-axis seamount distribution and its relationship with the segmentation of the East Pacific Rise and northern Pacific–Antarctic Ridge, 17°N–56°S

We used satellite-derived gravity anomaly maps and bathymetry data to analyze the distribution of off-axis seamounts on young crust on the flanks of the East Pacific Rise (EPR) and northern Pacific–Antarctic Ridge (PAR), from 17°N to 56°S latitude. We observed large-scale variations in the distribution of the volcanoes which we attribute to variations of the regional magmatic budget of the underlying mantle. Two distinct settings for off-axis volcanism were observed at the scale of second-order segments. The first one favors volcano growth near first- or second-order discontinuities. The second type occurs preferentially near the centers of second-order segments. We infer that the distribution of seamounts is controlled both by the availability of melt under the flanks of the ridge and by the vulnerability of the young lithosphere. Near the northern EPR and in some places near the PAR, the off-axis volcanism results mostly from fracturing of the crust near axial discontinuities, probably due to the thermomechanical stresses that develop in the cooling lithosphere. In the southern EPR, especially in the 14–19°S area, the young lithosphere is probably much more vulnerable to off-axis volcanism. The seamount distribution seems to reflect variations in the off-axis melt production at the scale of second-order segments (∼150 km). We infer that the difference in volcanic setting (close to segment ends or close to segment centers) between the northern and southern EPR results from the difference in spreading rate and, consequently, the variation in rheology of the young lithosphere.

A detailed geophysical study in the Canary Basin (eastern Atlantic): implications for the internal structure of 130 Ma oceanic crust

Summary We report the results of a detailed geophysical survey sited on two previously imaged dipping reflectors, one dipping towards and the other away from the present-day Mid-Atlantic Ridge axis, within Mesozoic-aged crust of the east Atlantic. The new data include gravity, magnetic and seismic refraction and reflection profiles collected in a grid that constrain the crustal structure as well as the reflector geometry. A second-order axial discontinuity, which is associated with anomalous crustal structure, is located within the survey area. The westward-dipping reflector can be traced from the Moho to the top of the igneous crust, where it coincides with a step of 800 m in the basement. This reflector is approximately planar, with a dip of 28°± 2° W and occurs ∼ 8 km north of the axial discontinuity. We interpret this reflector as being a whole-crustal normal fault that was generated within the median valley. The eastward-dipping reflector cannot be traced across seismic layer 2 and does not correlate with any feature of the basement. It penetrates 2–3 km into the upper mantle. Its reflection strength is about twice that of the Moho reflection. This reflector, which is approximately planar, strikes parallel to the isochrons, has a length of 18 ± 2 km and a dip of 28°± 2° E. We speculate that this reflector is also a fault that was generated on the western wall of the median valley but was ‘captured’ from the North American Plate by a lateral shift in the axis of spreading, possibly associated with a major change in plate motion.

Is the Moho flat? Seismic evidence for a rough crust-mantle interface beneath the northern Basin and Range

A rough crust-mantle boundary can produce amplitude versus offset characteristics similar to those of a second-order discontinuity (i.e., a gradient), namely a reduction in precritical PmP. Basin and Range reflection and refraction data can be simulated as well with rough Moho models as with gradient models. A rough Moho presents a different view of the interaction between the mantle and crust, where tectonic and magmatic structures are preserved and the crust and mantle are distinct. The rough Moho supports a crustal evolution model in which the crust is injected by mantle-derived mafic dikes and sills, but not to the extent that the lower crust is a continuous sequence of mafic/ultramafic intrusions, as commonly inferred from gradient models. As such, in the Basin and Range the Moho represents a well-defined chemical boundary.

Rapidly Visualizing Isophotes

For computer-aided design it is often necessary to identify regions of unwanted curvature in a surface. Isophotes can be used to detect first- and second-order discontinuities in a surface. This paper describes a method to visualize isophotes using existing hardware or software display and false colour capabilities. This method is faster and easier to implement than the traditional approach of explicitly computing isophotes.

Structural analysis of the segmentation of the Central Indian Ridge between 20�30?S and 25�30?S (Rodriguez Triple Junction)

The morphological characteristics of the segmentation of the Central Indian Ridge (CIR) from the Indian Ocean Triple Junction (25°30′S) to the Egeria Transform Fault system (20°30′S) are analyzed. The compilation of Sea Beam data from R/VSonne cruises SO43 and SO52, and R/VCharcot cruises Rodriguez 1 and 2 provides an almost continuous bathymetric coverage of a 450-km-long section of the ridge axis. The bathymetric data are combined with a GLORIA side-scan sonar swath to visualize the fabric of the ridge and complement the coverage in some areas. This section of the CIR has a full spreading rate of about 50 mm yr−1, increasing slightly from north to south. The morphology of the CIR is generally similar to that of a slow-spreading center, despite an intermediate spreading rate at these latitudes. The axis is marked by an axial valley 5–35 km wide and 500–1800 m deep, sometimes exhibiting a 100–600 m-high neovolcanic ridge. It is offset by only one 40km offset transform fault (at 22°40′S), and by nine second-order discontinuities, with offsets varying from 4 to 21 km, separating segments 28 to 85 km long. The bathymetry analysis and an empirical orthogonal function analysis performed on across-axis profiles reveal morphologic variations in the axis and the second-order discontinuities. The ridge axis deepens and the relief across the axial valley increases from north to south. The discontinuities observed south of 22°S all have morphologies similar to those of the slow-spreading Mid-Atlantic Ridge. North of 22°S, two discontinuities have map geometries that have not been observed previously on slow-spreading ridges. The axial valleys overlap, and their tips curve toward the adjacent segment. The overlap distance is 2 to 4 times greater than the offset. Based on these characteristics, these discontinuities resemble overlapping spreading centers (OSCs) described on the fast-spreading EPR. The evolution of one such discontinuity appears to decapitate a nearby segment, as observed for the evolution of some OSCs on the EPR. These morphological variations of the CIR axis may be explained by an increase in the crustal thickness in the north of the study area relative to the Triple Junction area. Variations in crustal thickness could be related to a broad bathymetric anomaly centered at 19°S, 65°E, which probably reflects the effect of the nearby Reunion hotspot, or an anomaly in the composition of the mantle beneath the ridge near 19°S. Other explanations for the morphological variations include the termination of the CIR at the Rodriguez Triple Junction or the kinematic evolution of the triple junction and its resultant lengthening of the CIR. These latter effects are more likely to account for the axial morphology near the Triple Junction than for the long-wavelength morphological variation.

Spatial and temporal variability of crustal magnetization of a slowly spreading ridge: Mid-Atlantic Ridge (20�?24� N)

The results of the two- and three-dimensional magnetic inversions performed on data located between 20°–24° N on the Mid-Atlantic Ridge indicate the crustal magnetization has decayed exponentially for the last 10 Ma, and that this decay has been fairly symmetric about the ridge axis. After removal of the mean temporal decay, the residual field is characterized by more positive magnetizations at the second-order discontinuities, regardless of initial magnetization direction. A model that involves the preferential emplacement of serpentinized lithologies near the discontinuities is proposed to explain this correlation. The temporal detrending method also indicates that several ridge-parallel depressions located on the flanks of the ridge axis are regions of more positive magnetizations. These bathymetric depressions may mark the locations of detachment faulting that occurred during amagmatic periods of extension. The general symmetry of the crustal magnetization about the ridge axis does not support the occurrence of continuous detachment faulting proposed to correspond to the inner and outer corners of ridge axis discontinuities.

The East Pacific Rise and its flanks 8?18� N: History of segmentation, propagation and spreading direction based on SeaMARC II and Sea Beam studies

SeaMARC II and Sea Beam bathymetric data are combined to create a chart of the East Pacific Rise (EPR) from 8°N to 18°N reaching at least 1 Ma onto the rise flanks in most places. Based on these data as well as SeaMARC II side scan sonar mosaics we offer the following observations and conclusions. The EPR is segmented by ridge axis discontinuities such that the average segment lengths in the area are 360 km for first-order segments, 140 km for second-order segments, 52 km for third-order segments, and 13 km for fourth-order segments. All three first-order discontinuities are transform faults. Where the rise axis is a bathymetric high, second-order discontinuities are overlapping spreading centers (OSCs), usually with a distinctive 3:1 overlap to offset ratio. The off-axis discordant zones created by the OSCs are V-shaped in plan view indicating along axis migration at rates of 40–100 mm yr−1. The discordant zones consist of discrete abandoned ridge tips and overlap basins within a broad wake of anomalously deep bathymetry and high crustal magnetization. The discordant zones indicate that OSCs have commenced at different times and have migrated in different directions. This rules out any linkage between OSCs and a hot spot reference frame. The spacing of abandoned ridges indicates a recurrence interval for ridge abandonment of 20,000–200,000 yrs for OSCs with an average interval of approximately 100,000 yrs. Where the rise axis is a bathymetric low, the only second-order discontinuity mapped is a right-stepping jog in the axial rift valley. The discordant zone consists of a V-shaped wake of elongated deeps and interlocking ridges, similar to the wakes of second-order discontinuities on slow-spreading ridges. At the second-order segment level, long segments tend to lengthen at the expense of neighboring shorter segments. This can be understood if segments can be approximated by cracks, because the propagation force at a crack tip is directly proportional to crack length.

PSV-wave propagation in a medium with local heterogeneities: a hybrid formulation and its application

SUMMARY Hybrid methods, which combine so-called ‘numerical’ and ‘analytical’ methods, are a successful tool for computing synthetic seismograms for media with localized heterogeneous regions. Each of the two methods is applied in that part of the medium in which it can be employed efficiently. This paper deals with a combination of a finite-difference algorithm, applied in the neighbourhood of the heterogeneity, and a frequency–wavenumber filter for the PSV-wave equation applied outside. This method is an alternative to the use of Green's theorem as a wavefield-continuation method (e.g. van den Berg 1984). In the hybrid method presented here, absorption can be incorporated additionally. For different models with first-and second-order discontinuities the hybrid method was tested successfully against the reflectivity method. The hybrid method was applied to study the effects of a heterogeneous lower crust on the wavefield. This application has been inspired by recent deep seismic reflection profiling, which often reveals a highly reflecting lower crust. The scattering models have been divided into three groups: horizontal, vertical and isotropic scatterers. The comparison of the reflection responses of these models shows that thin horizontal or small isotropic scatterers yield high reflected energy, coherent over a few kilometres, in the time window of lower crustal reflections. Vertical structures, characterized by a small correlation length in horizontal direction, have less energy in this time window, but they can generate a complex Moho reflection with a strong Moho coda. It is also shown that the acoustic approximation is mostly not sufficient to describe the wave propagation in such complex media.

Ridge axial segmentation in the Oman ophiolite: evidence from along-strike variations in the sheeted dyke complex

Abstract Evidence from along-strike variations in the sheeted dyke complex shows that the spreading axis at which the Oman ophiolite formed was segmented by first- and/or second-order discontinuities, most probably propagating rifts and overlapping spreading centres. We identify at least four, possibly up to seven, of these discontinuities along the c. 500 km ridge-parallel length of the ophiolite, but recognize no unequivocal transform fault structures. We show that the discontinuities in the sheeted dyke complex can be related to aspects of segmentation previously deduced from mantle flow fabrics and deep crustal structure, and conclude that the Oman palaeo-spreading axis was segmented on a scale of 50–100 km, comparable with that of a modern fast-spreading axis such as the East Pacific Rise.