Layer shortening and fold-shape development in the buckling of single layers

The progressive development of folds by buckling in single isolated viscous layers compressed parallel to the layering and embedded in a less viscous host is examined in several ways; by use of experiments, an analogue model to simulate simultaneous buckling and flattening and by an application of finite-element analysis.The appearance of folds with a characteristic wavelength in an initially flat layer occurs in the experiments for viscosity ratios (μ_layer/μ_host = μ₁/μ₂) of between 11 and 100; progressive fold development after the initial folds have appeared is similar in the experiments and in the finite-element models. Except for the finite-element model for μ₁/μ₂ = 1,000 layer-parallel shortening occurs in the early stages of folding and a stage is reached where little further changes in arc length occur. The amount of layer-parallel shortening increases with decreasing viscosity contrast, and becomes relatively unimportant after the folds have attained limb dips of about 15°–25°.Thickness variations with dip are only significant here for the finite-element model with μ₁/μ₂ = 10, and in experiments for μ₁/μ₂ = 5 where the layer is initially in the form of a moderate-amplitude sine wave. The variations range from a parallel to a near-similar fold geometry, and in general depend on the viscosity contrast, the degree of shortening and the initial wavelength/thickness ratio. They are very similar to the variations predicted by the analogue model of combined buckling and flattening. The difference between the thickness/dip variations in a fold produced by buckling at low viscosity contrast and one produced by flattening a parallel fold is marked at high limb dips and very slight at low limb dips.Many natural folds in isolated rock layers or veins show thickness/dip relationships expected for a flattened parallel fold, and some show relationships expected for buckling at low viscosity contrasts. Studies of the wavelength/thickness ratios in natural folds have suggested that competence contrast is often low. Many folds in isolated rock layers or veins whose geometry may vary between parallel and almost similar, and may be indistinguishable from those of flattened parallel folds, have probably developed by a process of buckling at low viscosity contrasts.     

The structural evolution of sheath folds: A case study from Cap de Creus

It has long been recognised that within zones of intense non-coaxial deformation, fold hinges may rotate progressively towards the transport direction ultimately resulting in highly curvilinear sheath folds. However, there is a surprising lack of detailed and systematic field analysis of such “evolving” sheath folds. This case study therefore focuses on the sequential development of cm-scale curvilinear folds in the greenschist-facies El Llimac shear zone, Cap de Creus, Spain. This simple shear-dominated dextral shear zone displays superb three dimensional exposures of sheath folds defined by mylonitic quartz bands within phyllonite. Increasing amounts of fold hinge curvature (δ) are marked by hinge segments rotating into sub-parallelism with the mineral lineation (Lm), whilst the acute angle between the axial-planar hinge girdle and foliation (ω) also displays a sequential reduction. Although Lm bisects the noses of sheath folds, it is also clearly folded and wrapped-around the sheath hinges. Lm typically preserves a larger angle (θ) with the fold hinge on the lower limb (L) compared to the upper (U) limb (θL > θU), suggesting that Lm failed to achieve a steady orientation on the lower limb. Adjacent sheath fold hinges forming fold pairs may display the same sense of hinge arcing to define synthetic curvature, or alternatively opposing directions of antithetic curvature. Such patterns reflect original buckle fold geometries coupled with the direction of shearing. The ratio of long/short fold limbs decreases with increasing hinge curvilinearity, indicating sheath folds developed via stretching of the short limb, rather than migrating or rolling hinge models. This study unequivocally demonstrates that both hinges of fold pairs become curvilinear with sheaths closing in the transport direction recording greater hinge-line curvilinearity compared to adjacent return hinges. This may provide a useful guide to bulk shear sense.     

Kinetics of high-pressure phase transformations: Implications to the evolution of the olivine → spinel transition in the downgoing lithosphere and its consequences on the dynamics of the mantle

The rate of a high-pressure phase transition increases exponentially with temperature (T) and overpressure or pressure beyond equilibrium (ΔP). It is also greatly promoted by introducing shear stress, diminishing grain size, and adding water or other catalysts to the reactants. For an isothermal and isobaric transition with no compositional change, if steady state of nucleation on grain surfaces is attained, the rate equation can be expressed: (1) before site saturation by: X = 1 − exp(−Kt⁴), where

Full-size image (2K)

and (2) after site saturation by: X = 1 − exp(−K′T), where

Full-size image (1K)

, where X is volume fraction of completion of transformation, t is time, and the C's are characteristic constants. C₁ and C₉ are functions of grain size, C₃ and C₆ are functions of shear stress. All the C's are almost independent of temperature and pressure. Thus, if X as a function of T, ΔP, and t over a narrow P-T range can be experimentally determined, the C's can be calculated and the effect of grain size and shear stress on the rate of transformation can be evaluated. The isothermal and isobaric rate equations for a given composition, shear stress, and grain size are then experimentally determinable. The non-isothermal and non-isobaric rate equation can be calculated from the isothermal and isobaric ones if the rate of penetration into the metastability field is known. The important feature of the kinetics of high-pressure phase transitions predicted by these rate equations is that for a given rate of penetration into the metastability field, there can be defined a characteristic temperature, T_ch, below which the rate of the transition is virtually zero no matter how metastable the material is. For the olivine → spinel transition in the mantle, this characteristic temperature may be as high as 700° C. Thus, in a fast moving downgoing slab, the temperature at its cold center may remain below T_ch even down to depths in excess of 600 km, thereby greatly depressing the olivine—spinel phase boundary.At an early stage in the development of a downgoing slab, the plunging speed is slow. This allows the interior of the slab to heat up and the olivine → spinel transition to proceed rapidly and near equilibrium. As a result, the olivine—spinel phase boundary in the slab will be distorted upwards. The rising of the denser spinel phase then provides an additional driving force which accelerates the plate. Since the upper portion of the slab is pulled from below and the lower portion pushed from above, earthquakes of down-dip extension will occur in the upper mantle while those of down-dip compression will originate in the transition zone. Because the transformation occurs close to equilibrium, there will be an aseismic region separating the two seismic zones. When the plate velocity exceeds a certain limit, the temperature in the cold interior becomes low enough to depress the olivine → spinel transition. The phase boundary is then distorted downwards. The buoyant force thereby created will reduce the driving force, and the plunging speed of the plate will approach a steady state. In addition, the buoyant force will compress the slab from below and result in earthquakes of down-dip compression throughout the length of the slab. Now the olivine → spinel transition is so far from equilibrium that the reaction becomes implosive. A rise in frequency of deep earthquakes towards the implosion region in the lower transition zone is thus predicted. Therefore, as well as stabilizing the plate velocity, the olivine → spinel transition may also control earthquake distributions throughout the downgoing slab.     

Oxygen isotopic fractionation during inorganic calcite precipitation ― Effects of temperature, precipitation rate and pH

Stable oxygen isotopic fractionation during inorganic calcite precipitation was experimentally studied by spontaneous precipitation at various pH (8.3 < pH < 10.5), precipitation rates (1.8 < log R < 4.4 μmol m^− 2 h^− 1) and temperatures (5, 25, and 40 °C) using the CO₂ diffusion technique.The results show that the apparent stable oxygen isotopic fractionation factor between calcite and water (α_{calcite–water}) is affected by temperature, the pH of the solution, and the precipitation rate of calcite. Isotopic equilibrium is not maintained during spontaneous precipitation from the solution. Under isotopic non-equilibrium conditions, at a constant temperature and precipitation rate, apparent 1000lnα_{calcite–water} decreases with increasing pH of the solution. If the temperature and pH are held constant, apparent 1000lnα_{calcite–water} values decrease with elevated precipitation rates of calcite. At pH = 8.3, oxygen isotopic fractionation between inorganically precipitated calcite and water as a function of the precipitation rate (R) can be described by the expressions

at 5, 25, and 40 °C, respectively.The impact of precipitation rate on 1000lnα_{calcite–water} value in our experiments clearly indicates a kinetic effect on oxygen isotopic fractionation during calcite precipitation from aqueous solution, even if calcite precipitated slowly from aqueous solution at the given temperature range. Our results support Coplen's work [Coplen T. B. (2007) Calibration of the calcite–water oxygen isotope geothermometer at Devils Hole, Nevada, a natural laboratory. Geochim. Cosmochim. Acta 71, 3948–3957], which indicates that the equilibrium oxygen isotopic fractionation factor might be greater than the commonly accepted value.     

Magnesite growth rates as a function of temperature and saturation state

Magnesite growth rates and step velocities have been measured systematically as a function of temperature from 80 to 105 °C and saturation state in 0.1 M NaCl solutions using hydrothermal atomic force microscopy (HAFM). The observations indicate that at these conditions magnesite precipitation is dominated by the coupling of step generation via spiral growth at screw dislocations and step advancement away from these dislocations. As these two processes occur in series the slowest of these dominates precipitation rates. At 100 °C magnesite growth rates (r) determined by HAFM are consistent with

r=k(Ω-1)²,

where k is a constant equal to 6.5 × 10⁻¹⁶ mol/cm²/s and Ω is the saturation index with respect to magnesite. This equation is consistent with spiral growth step generation controlling magnesite precipitation rates. Corresponding magnesite precipitation rates measured using mixed-flow reactors are shown to be consistent with both the rates measured by HAFM and the spiral growth theory, confirming the rate limiting mechanism. Step advancement, however, is observed to slow far faster than step generation with decreasing temperature; the activation energy for step advancement is 159 kJ/mol whereas step generation rates have an estimated activation energy of 60 kJ/mol. As such, it seems likely that at ambient temperatures magnesite growth is limited by very slow step advancement rates.     

Solubility and dissolution rate of silica in acid fluoride solutions

We performed 57 batch reactor experiments in acidic fluoride solutions to measure the dissolution rate of quartz. These rate data along with rate data from published studies were fit using multiple linear regression to produce the following non-unique rate law for quartz

where 10^−5.13 < a_HF < 10^1.60, −0.28 < pH < 7.18, and 298 < T < 373 K. Similarly, 97 amorphous silica dissolution rate data from published studies were fit by multiple linear regression to produce the following non-unique rate law for amorphous silica

where 10^−2.37 < a_HF < 10^1.61, −0.32 < pH < 4.76 and 296 < T < 343 K. Regression of the rates versus other combinations of solution species, e.g.  + H⁺, F⁻ + H⁺, HF + , HF + F⁻, or  + F⁻, produced equally good fits. Any of these rate laws can be interpreted to mean that the rate-determining step for silica dissolution in fluoride solutions involves a coordinated attack of a Lewis acid, on the bridging O atom and a Lewis base on the Si atom. This allows a redistribution of electrons from the Si–O bond to form a O–H group and a Si–FH group.     

Multicomponent magnetization in western Dolpo (Tethyan Himalaya, Nepal): tectonic implications

A palaeomagnetic study has been carried out in the Tethyan Himalaya (TH; the northern margin of Greater India). Twenty-six palaeomagnetic sites have been sampled in Triassic low-grade metasediments of western Dolpo. Two remanent components have been identified. A pyrrhotite component, characterized by unblocking temperatures of 270–335 °C, yields an in situ mean direction of D=191.7°, I=−30.9° (k=29.5, α₉₅=5.7°, N=23 sites). The component fails the fold test at the 99% confidence level (k_{in situ}/k_bed=6.9) and is therefore of postfolding origin. For reason of the low metamorphic grade, this pyrrhotite magnetization is believed to be of thermo-chemical origin. Geochronological data and inclination matching indicate an acquisition age around 35 Ma. The second remanence component has higher unblocking temperatures (>400 °C and up to 500–580 °C range) and resides in magnetite. A positive fold test and comparison with expected Triassic palaeomagnetic directions suggest a primary origin.The postfolding character of the pyrrhotite component, and its interpreted age of remanence acquisition, implies that the main Himalayan folding is older than 35 Ma in the western Dolpo area. This study also suggests that the second metamorphic event (Neo-Himalayan) was more significant in the Dolpo area than the first (Eo-Himalayan) one.A clockwise rotation of 10–15° is inferred from the pyrrhotite component, which is compatible with oroclinal bending and/or rotational underthrusting models. This rotation is also supported by the magnetite component, indicating that no rotation of the Tethyan Himalaya relative to India took place before 35 Ma.     

Finite-element folds of similar geometry

Model folds of similar geometry have been produced by using the finite-element method and the constitutive relations of a layer of wet quartzite embedded in a marble matrix with an initially sinusoidal configuration and a 10° limb dip. The power law for steady-state flow of Yule Marble (Heard and Raleigh, 1972) is used for the matrix and our new law for Canyon Creek quartzite deformed in the presence of water is used for the layer. The equiv- alent viscosity of the wet quartzite is highly temperature-sensitive, giving rise to a strong temperature dependence of the quartzite: marble viscosity ratio which, at a strain rate of 10⁻¹⁴/sec, drops from 543 at 200° to 0.13 at 800°C. At 375°C (η_q/η_m = 10), concentric folds develop at all strains to 80% natural shortening and stress, finite strain and viscosity distributions are somewhat similar to those found previously. Raising the temperature to 550° C (η_q/η_m = 1), at any stage of prior amplification, causes the folds to flatten with increasing strain, accompanied by attenuation of limbs and thickening of hinges, leading to folds with similar geometries and isoclinal folds at extreme strains. The effects are more pronounced at higher temperatures and at 700° C (η_q/η_m = 0.3) limb attenuation is so severe as to give rise to unrealistic geometries. At temperatures below about 600° C (η_q/η_m = 2), similar folds do not form. It thus appears as if a viscosity contrast near unity is required to produce similar folds in rocks, under the conditions simulated and different temperature dependencies of viscosities of materials in layered sequences is one important means of reducing viscosity contrasts.     

Diffusion of helium in zircon and apatite

Diffusion of helium has been characterized in natural zircon and apatite. Polished slabs of zircon and apatite, oriented either normal or parallel to c were implanted with 100 keV ³He at a dose of 5 × 10^{15 3} He/cm². Diffusion experiments on implanted zircon and apatite were run in Pt capsules in 1-atm furnaces. ³He distributions following experiments were measured with Nuclear Reaction Analysis using the reaction ³He(d,p)⁴He. For diffusion in zircon we obtain the following Arrhenius relations:

Although activation energies for diffusion normal and parallel to c are comparable, there is marked diffusional anisotropy, with diffusion parallel to c nearly 2 orders of magnitude faster than transport normal to c. These diffusivities bracket the range of values determined for He diffusion in zircon in bulk-release experiments, although the role of anisotropy could not be directly evaluated in those measurements.In apatite, the following Arrhenius relation was obtained over the temperature range of 148–449 °C for diffusion normal to c:

In contrast to zircon, apatite shows little evidence of anisotropy. He diffusivities obtained in this study fall about an order of magnitude lower than diffusivities measured through bulk release of He through step-heating, and within an order of magnitude of determinations where ion implantation was used to introduce helium and He distributions measured with elastic recoil detection.Since the diffusion of He in zircon exhibits such pronounced anisotropy, helium diffusional loss and closure cannot be modeled with simple spherical geometries and the assumption of isotropic diffusion. A finite-element code (CYLMOD) has recently been created to simulate diffusion in cylindrical geometry with differing radial and axial diffusion coefficients. We present some applications of the code in evaluating helium lost from zircon grains as a function of grain size and length to diameter ratios, and consider the effects of “shape anisotropy”, where diffusion is isotropic (as in the case of apatite) but shapes of crystal grains or fragments may depart significantly from spherical geometry.     

Relationships between joint apparent separation, Schmidt hammer rebound value, and distance to faults, in rocky outcrops, Calabria, Southern Italy

In order to investigate how the apparent separation of jointing varies according to the distance from faults, and how the mechanical properties of rock masses depend on this distance and jointing density, a number of regression analysis were performed on the variables s (apparent joint separation), d (distance from major fault), and sh (rebound value of Schmidt hammer).The variables were measured at 380 stations distributed over a wide study area located in the Aspromonte range in Calabria, Southern Italy.The most significant results of simple regression analysis are expressed by the formulas:

s=c+Fd⁰⁵