Simulation of normal distributed smooth fields by Karhunen-Loéve expansion in combination with kriging

Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.     

Influence of the zone of weakness on dip angle and shear heating of subducted slabs

We study the importance of the zones of weakness and the pattern of downgoing flow in steady-state models of subducting lithosphere, which interacts mechanically and thermally with the ambient mantle. The non-linear system of governing equations consists of (i) the momentum equation in stream function formulation and (ii) the steady-state heat transfer equation including conduction and advection of heat and dissipation. A finite element method has been applied to this system. We consider the viscosity to be a non-linear function of both the temperature and the stream function. In steady-state two-dimensional (2D) flow, the stream function isolines follow material trajectories. They are used to follow the top of the subducting slab, which because of its possible increase in water content, is assumed to have a lower viscosity. The zone of weakness has been thus obtained in the self-consistent fashion since the stream function as well as the temperature are the output from our modeling and no a priori assumptions about the shape of the bending lithosphere are taken into account. It was shown that several orders decrease of viscosity in the zone of weakness is required to obtain the dip angle of about 45°. If the decrease of viscosity is not sufficient enough, the subducted slab either sinks almost vertically or does not exhibit a plate-like behavior. We have also demonstrated that shear heating can unrealistically increase at the zone of weakness for fast subductions if decrease of viscosity is underestimated.     

Plastic flow strength of jadeite and diopside investigated by microindentation hardness tests

Microindentation hardness tests were performed on jadeite and diopside, being end members of the omphacite solid solution series. At temperatures between 300 and 750 °C, the hardness of jadeite ranges from 7.4 to 8.5 GPa, that of diopside from 4.9 to 6.1 GPa. Jadeite is significantly stronger than diopside in the low-temperature plasticity regime. Normalization of the hardness-derived yield stress with respect to the shear modulus considerably reduces the strength contrast. The normalized Peierls stress is identical for jadeite and diopside. This indicates that jadeite and diopside belong to the same isomechanical group. The hardness-derived yield stress for jadeite as well as for diopside is used to estimate flow law parameters for the low-temperature plasticity regime.     

On the effect of a low viscosity asthenosphere on the temporal change of the geoid—A challenge for future gravity missions

New satellite technology to measure changes in the Earth’s gravity field gives new possibilities to detect layers of low viscosity inside the Earth. We used density models for the Earth mantle based on slab history as well as on tomography and fitted the viscosity by comparison of predicted gravity to the new CHAMP gravity model. We first confirm that the fit to the observed geoid is insensitive to the presence of a low viscosity anomaly in the upper mantle as long as the layer is thin ( 200 km) and the viscosity reduction is less than two orders of magnitude. Then we investigated the temporal change in geoid by comparing two stages of slablet sinking based on subduction history or by advection of tomography derived densities and compared the spectra of the geoid change for cases with and without a low viscosity layer, but about equal fit to the observed geoid. The presence of a low viscosity layer causes relaxation at smaller wavelength and thus leads to a spectrum with relatively stronger power in higher modes and a peak around degrees 5 and 6. Comparing the spectra to the expected degree resolution for GRACE data for a 5 years mission duration shows a weak possibility to detect changes in the Earth’s gravity field due to large scale mantle circulation, provided that other causes of geoid changes can be taken into account with sufficient accuracy. A discrimination between the two viscosity cases, however, demands a new generation of gravity field observing satellites.     

Continuous Multi‐probe Measurements in Fastflowing Waters Using a Hydrographic Slot

Taking continuous spatiotemporal in situ measurements with multi‐probes in fast‐flowing waters/rivers can be problematic because the sensors may be damaged by high shear forces and flotsam. To protect the multi‐probe and to enable easy access for the maintenance and calibration of the sensors, a special multi‐probe holder fixed in a hydrographic slot was developed. The validation of the probe system revealed a “memory effect” at short time scales (< 10 s) within sharp gradients caused by the overflow container of the multi‐probe rack keeping the sensors submerged in the sample water. Continuously recorded data (conductivity, temperature, pH, oxygen concentration and saturation, as well as in vivo fluorescence of chlorophyll‐a) from a research cruise on board the RV ALBIS along the river Elbe (river km 309) and entering the river Saale are presented. This river stretch upstream of the city of Magdeburg to the mouth of the Saale tributary was found to have a complex physicochemical character, which is attributable to the long mixing process of water from the river Saale and the river Elbe.