Three dimensional application of the complementary mild-slope equation

The complementary mild-slope equation (CMSE) is a depth-integrated equation, which models refraction and diffraction of linear time-harmonic water waves. For 2D problems, it was shown to give better agreements with exact linear theory compared to other mild-slope (MS) type equations. However, no reference was given to 3D problems. In contrast to other MS-type models, the CMSE is derived in terms of a stream function vector rather than in terms of a velocity potential. For the 3D case, this complicates the governing equation and creates difficulties in formulating an adequate number of boundary conditions. In this paper, the CMSE is re-derived using Hamilton's principle from the Irrotational Green–Naghdi equations with a correction for the 3D case. A parabolic version of it is presented as well. The additional boundary conditions needed for 3D problems are constructed using the irrotationality condition. The CMSE is compared with an analytical solution and wave tank experiments for 3D problems. The results show very good agreement.     

A Continuum Damage–Breakage Faulting Model and Solid-Granular Transitions

We present a thermodynamically-based formulation for mechanical modeling of faulting processes in the seismogenic brittle crust using a continuum damage–breakage rheology. The model combines previous results of a continuum damage framework for brittle solids with continuum breakage mechanics for granular flow. The formulation accounts for the density of distributed cracking and other internal flaws in damaged rocks with a scalar damage parameter, and addresses the grain size distribution of a granular phase in a failure slip zone with a breakage parameter. The stress–strain relation and kinetics of the damage and breakage processes are governed by the total energy function of the system, which combines the energy of the damaged solid with the energy of the granular material. A dynamic brittle instability is associated with a critical level of damage in the solid, leading to loss of convexity of the solid energy function and transition to a granular phase associated with lower energy level. A non-local formulation provides an intrinsic length scale associated with the internal damage structure, which leads to a finite length scale for damage localization that eliminates the unrealistic singular localization of local models. Shear heating during deformation can lead to a secondary finite-width internal localization. The formulation provides a framework for studying multiple aspects of brittle deformation, including potential feedback between evolving elastic moduli and properties of the slip localization zone and subsequent rupture behavior. The model has a more general transition from slow deformation to dynamic rupture than that associated with frictional sliding on a single pre-existing failure zone, and gives time and length scales for the onset of the dynamic fracturing process. Several features including the existence of finite localization width and transition from slow to rapid dynamic slip are illustrated using numerical simulations. A configuration having an existing narrow slip zone with localized damage produces for appropriate loading conditions an overall cyclic stick–slip motion. The simulated frictional response includes transitions from friction coefficient of ~0.7 at low slip velocity to dynamic friction below 0.4 at slip rates above ~0.1 m/s, followed by rapidly increasing friction for slip rates above ~1 m/s, consistent with laboratory observations.     

Evolution and degradation of flat‐top mesas in the hyper‐arid Negev,Israel revealed from 10Be cosmogenic nuclides

Mesas are ubiquitous landforms in arid and semiarid regions and are often characterized by horizontal stratified erodible rocks capped by more resistant strata. The accepted conceptual model for mesa evolution and degradation considers reduction in the width of the mesa flat‐top plateau due to cliff retreat but ignores possible denudation of the mesa flat‐top and the rates and mechanism of erosion. In this study we examine mesas in the northeastern hyperarid Negev Desert where they appear in various sizes and morphologies and represent different stages of mesa evolution. The variety of mesas within a single climatic zone allows examination of the process of mesa evolution through time. Two of the four sites examined are characterized by a relatively wide (200–230 m) flat‐top and a thick caprock whereas the other two are characterized by a much narrower remnant flat‐top (several meters) and thinner caprock. We use the concentration of the cosmogenic nuclide ¹⁰Be for: (a) determining the chronology of the various geomorphic features associated with the mesa; and (b) understanding geomorphic processes forming the mesa. The ¹⁰Be data, combined with field observations, suggest a correlation between the width of flat‐top mesa and the denudation and cliff retreat rates. Our results demonstrate that: (a) cliff retreat rates decrease with decreasing width of the flat‐top mesa; (b) vertical denudation rates increase with decreasing width of the flat‐top mesa below a critical value (~60 m, for the Negev Desert); (c) the reduction in the width of the flat‐top mesa is driven mainly by cliff retreat accompanied by extremely slow vertical denudation rate which can persist for a very long time (>10⁶ Ma); and (d) when the width of the mesa decreases below a certain threshold, its rate of denudation increases dramatically and mesa degradation is completed in a short time. Copyright © 2014 John Wiley & Sons, Ltd.     

Stratigraphic Framework and Heavy Minerals of the Continental Shelf of Onslow and Long Bays,North Carolina

One hundred fourteen vibracores from the Atlantic continental shelf offshore of southeastern North Carolina were opened, described, and processed over several contract years (years 6-9) of the Minerals Management Service Association of American State Geologists Continental Margins program. Reports for years 9 and 10 of the program compiled the results of the work and assembled the data for release as an interactive CD-ROM report, respectively. The continental shelf of Onslow and Long Bays consists predominantly of outcropping Cretaceous through late Tertiary geologic units. Nearshore these units are covered and incised by late Tertiary and Quaternary units. From oldest to youngest, formally recognized geologic units mapped as part of this study are the Late Cretaceous Peedee Formation a muddy, fine-to medium-grained quartz sand with trace amounts of glauconite and phosphate; the Paleocene Beaufort Forma tion a muddy, fine-to medium-grained glauconitic quartz sand with locally occurring turritelid-mold biosparrudite; the middle Eocene Castle Hayne Forma tion a sandy bryozoan biomicrudite and biosparrudite; the Oligocene River Bend Formation a sandy molluscan-mold biosparrudite; and the Miocene Pungo River Formation a medium-grained, poorly sorted slightly shelly phosphatic sand. Infor mal units include a very widespread, unnamed fine-to very fine grained, well-sorted, dolomitic muddy quartz sand that is biostratigraphically equivalent to the Oligocene River Bend Formation; several large valley-fill lithosomes composed of biomicrudite, biomicrite, and biosparrudite of Plio Pleistocene age; muddy, shelly sands and silty clays of Pliocene, Pleistocene, or mixed Plio Pleistocene age; and loose, slightly shelly, medium- to coarse-grained sands assigned a Holocene age. Heavy minerals (SG>2.96) comprise an average of 0.54 wt % (on a bulk-sam ple basis) of the sediments in 306 samples derived from the 114 vibracores. Heavy-mineral content ranges from <0.01 to 3.69 wt %. The economic heavy mineral content (EHM ilmenite zircon rutile aluminosilicates leucoxene [altered ilmenite] monazite) of the bulk samples averages 0.26 wt % in a range of <0.01-1.70 wt %. As a percentage of the heavy-mineral concentrate, the average EHM value is 45.78 % in a range of 0.27-68.60 %. The distribution of heavy minerals offshore of southeastern North Carolina is controlled by the lithostratigraphic framework. The unnamed Oligocene sand unit has the highest heavy-mineral content, averaging 0.86 wt % on a bulk-sample basis. The remaining geologic units and their heavy-mineral content (in decreasing order of abundance) are Beaufort (0.64 %), Holocene sand (0.60 %), Plio-Pleistocene muddy sand and silty clay (0.59 %), Peedee (0.42 %), River Bend (0.34 %), Plio-Pleistocene carbonate (0.12 %), and Castle Hayne (0.08 %). The heavy-mineral assemblage is fairly consistent throughout the different units. Significantly smaller percentages of heavy minerals correlate with increased amounts of CaCO3 in the sediments. The sediments analyzed in this study have significantly lower overall heavymineral content, as well as lower EHM content than sediments that are known to host commercially important heavy-mineral deposits in the southeastern United States. The potential for economic deposits of heavy minerals in the area of this study, therefore, appears to be limited.     

Pan-glacial—a third state in the climate system

Radiative energy-balance models reveal that Earth could exist in any one of three discrete climate states—'non-glacial' (no continental ice-sheets), 'glacial-interglacial' (high-latitude ice-sheets) or 'pan-glacial' (ice-sheets at all latitudes)—yet only the first two were represented in Phanerozoic time. There is mounting evidence that pan-glacial states existed at least twice in the Cryogenian (roughly 750–635 Ma), the penultimate period of the Neoproterozoic. Consensus is lacking on whether the world ocean was fully glaciated ('snowball' model) or largely unglaciated ('slushball' model). The first appearances of multicellular animal fossils (diapause eggs and embryos in China, and sponge-specific biomarkers in Oman), being closely associated with the last pan-glacial state, revive speculation that environmental forces had a hand in the origin of metazoa.     

Gusev photometric variability as seen from orbit by HRSC/Mars-express

Minnaert and two-term phase function Hapke models are used to describe the photometric properties of the martian surface using HRSC (High Resolution Stereo Camera) multi-angular observations acquired along the ongoing Mars-Express mission. These observations can be pieced together to derive integrated phase functions over a wide range of phase angles. The photometric diversity at 675 nm, as seen from orbit, of the martian surface properties across Gusev is depicted with seven units. Three photometric units widespread across the flanks of Apollinaris Patera flank and the floor of Gusev Crater are identified as having high single scattering albedo with rather forward scattering properties, low to intermediate macroscopic roughness and porous or not compacted powdered surface state as indicated by the opposition parameters. Another unit has the highest single scattering albedo, the smoothest surface in terms of macroscopic roughness, associated with an extremely forward scattering behavior. The opposition parameters are consistent with the presence of transparent particles in the surface powder layer. The distribution of this unit appears quite intermittent across the crater and does not seem to indicate any relationship with a given morphological structure. It may correspond to sparse areas where the structure of the surface dust layer is the most preserved. The most pronounced photometric changes are observed in three units associated with the low-albedo features corresponding to dark wind streaks. These units have a low single scattering albedo, are the most backscattering surfaces across Gusev, have a high surface roughness and present variable surface states as shown by the opposition parameters estimates, consistent with the occurrence of large grains organized in more or less packed layers. Clear differences are seen among these units in terms of opposition effect. While one exhibits typical characteristics for the opposition effect, another appears more unusual in terms of lobe width and the third suggests the occurrence of a packed/compressed/narrow size distribution powder surface. The opposition effect thus appears to play a significant role suggesting that the surface state optical properties across Gusev are strongly influenced by the porosity and packing characteristics or grain size distribution of the upper layer of the martian soil. The mapping aspect of the investigation is quite useful to get a better sense of the meaning of the observed photometric variations. Indeed, the Hapke modeling suggests that surface organization (surface roughness, packing state) is more important than the simple physical characterization of the intrinsic optical properties of the constitutive particles. Given the overall spatial patterns derived from the photometric analysis, the variations, at least for the western and central part of Gusev Crater, are likely partly driven by the prevailing wind regimes, considered to be oriented north-northwest/south-southeast and disturbing the very upper surface layer. The present photometric results agree with independent investigations based on thermal inertia, reflectance spectroscopy, in situ photometric and microscopic imaging and support the idea of a thin layer of fine-grained dust, being stripped off in the low albedo units to reveal a dark basaltic substrate comprising coarse-grained materials.