Lagrangian coherent structures,transport and chaotic mixing in simple kinematic ocean models

Methods of dynamical system’s theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean—a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle’s scattering and transport. A numerical analysis reveals a non-attracting chaotic invariant set Λ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle’s coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time. The clusters appear due to existence of regions of stability in the phase space which is the physical space in the advection problem.     

Spin-polarized transport in structures with tunnel barriers

We study spin-dependent transport theoretically in structures with a barrier composed of magnesium oxide and ferromagnetic electrodes with a noncollinear orientation of magnetizations in the ferromagnetic layers. We consider the effects of tunnel magnetoresistance and of the spin torque in the two-band s-d model using the Keldysh technique of nonequilibrium Green’s functions with the s-d hybridization taken into account. We investigate the role of a chromium layer as an additional spin filter for s electrons and the possibility of increasing the effects of tunnel magnetoresistance and of the spin torque for the resonance tunneling in the five-layer structure Fe|Cr|Fe|MgO|Fe.     

Viscoelastic characteristics of the interaction between structures and soils

The shear coefficients of a body relative to the soil and the shear creep and relaxation kernels are determined from the data of creep and relaxation tests.Urazbaev Institute of Mechanics and Earthquake Resistance of Structures, Academy of Sciences of the SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 207–211, March–April, 1973.     

Measurement and comparison of soil structures.

A model is established to describe the structures of tilled soils using Markov chain theory. The effectiveness of the model in describing soil structures, and its accuracy when the model parameters are determined from limited field data is investigated by a consideration of variances of the transition probabilities and Markov chain state occurances in finite length chains. Criteria for correlation of soil structures at small horizontal and vertical displacements are derived, in order to establish distances at which soil structures become effectively independent. In this, a mathematical analysis is made of limiting covariances, generally applicable to the type of Markov chain used in describing these structures, in order to drastically reduce computing time in processing field data. Similarity coefficients are defined from the theory to measure similarity in different soil structures, and are compared in practice.     

Classroom-based interventions in mathematics education: relevance, significance, and applicability

This special issue discusses various pedagogical innovations and myriad of significant findings. This commentary is not a synthesis of these contributions, but a summary of my own reflections on selected aspects of the nine papers comprising the special issue. Four themes subsume these reflections: (1) Gestural Communication (Alibali, Nathan, Church, Wolfgram, Kim and Knuth 2013); (2) Development of Ways of Thinking (Jahnke and Wambach 2013; Lehrer, Kobiela and Weinberg 2013; Mariotti 2013; Roberts and A. Stylianides 2013; Shilling-Traina and G. Stylianides 2013; Tabach, Hershkowitz and Dreyfus 2013); (3) Learning Mathematics through Representation (Saxe, Diakow and Gearhart 2013); and (4) Challenges in Dialogic Teaching (Ruthven and Hofmann 2013).     

A Green's function for the energy transport equation in Oseen flow

Zusammenfassung Es werden mehrere Darstellungen in geschlossener Form für die Greensche Funktion der Energieübertragungsgleichung einer achsensymmetrischen Oseenschen Strömung gegeben.     

Parametrization for mesoscale ocean transport through random flow models

We describe a mathematical approach based on homogenization theory toward representing the effects of mesoscale coherent structures on large-scale transport in the ocean. We demonstrate the approach on a deterministic and a random model flow. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adapted complex structures and the geodesic flow

In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the “complexifier” approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the “imaginary-time geodesic flow” to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by “composition with the imaginary-time geodesic flow.” We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.     

Method for calculation of flow distribution in a gas transport system

A method is proposed for iterative refinement of hydraulic resistances of portions of a high-pressure gas transport system with consideration of its current state. The algorithm is realized as a set of computer programs. Numerical experiments performed with data from a functioning interindustry system show good agreement between calculated values and dispatcher data.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 98–102, 1989.     

A new multiscale model for flow and transport in unconventional shale oil reservoirs

A new multiscale computational model for two-phase gas/oil flow in a multiporosity shale formation composed of three levels of porosity associated with nano/micro pores and hydraulic fractures is rigorously constructed within the framework of the homogenization procedure in conjunction with the Discrete Fracture Modeling, where fractures are treated as (n-1) interfaces (n=2, 3) immersed in the domain occupied by the matrix. Effective equations are obtained by upscaling the microstructural information of the shale oil formation with matrix composed of three distinct solid phases: the organic matter, constituted by kerogen aggregates containing particles and nanopores with adsorbed gas, and the pyrobitumen network, also containing an organic solid with micropores filled by tight oil and dissolved gas, along with the inorganic solid composed of clay, quartz and calcite, assumed impermeable and free of adsorption. Such distinct solid phases are separated from each other by the network of interparticle pores. Together with the pyrobitumen such a network form the pathways for multiphase flow in the matrix whereas the kerogen aggregates are treated as disconnected inclusions playing the role of storage sites for adsorbed gas. The homogenization of the multiphase flow model of black oil type gives rise to new pressure and saturation equations with effective coefficients strongly correlated with the shale microstructure, volume fractions and total organic content (TOC). Constitutive laws for the effective hydraulic conductivity and retardation parameter, which captures adsorption of methane in the nanopores, are numerically reconstructed by solving the local cell problems arising from the homogenization procedure. In particular the partition coefficient is computed from adsorption isotherms rigorously constructed within the framework of the Thermodynamics of confined gases seated on the Density Functional Theory (DFT). The effective equations in the matrix resemble in form of a generalized black oil model coupled with the two-phase flow model posed in the subdomain occupied by the network of hydraulic fractures. A macroscopic model is obtained by averaging the mass conservation equation across the fracture aperture giving rise to reduced balance laws posed in a network of reduced (n-1)-dimension (n=2, 3) supplemented by a source term arising from the jump in the oil/gas fluxes in the shale matrix. The resultant coupled Discrete Fracture/Multiscale model consists of a first attempt at constructing a rigorous correlation between the nature of the macroscopic multiphase flow equations and the local shale-microstructure mainly characterized by the simultaneous presence of inorganic and organic matters with the latter containing nanopores. Numerical simulations of gas/oil withdrawal are performed to accurately predict hydrocarbon movement in stimulated shale oil formations.     

On characterization of Poisson and Jacobi structures

We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.     

Computational homogenization of non-stationary transport processes in masonry structures

A fully coupled transient heat and moisture transport in a masonry structure is examined in this paper. Supported by several successful applications in civil engineering the nonlinear diffusion model proposed by Künzel (1997) [16] is adopted in the present study. A strong material heterogeneity together with a significant dependence of the model parameters on initial conditions as well as the gradients of heat and moisture fields vindicates the use of a hierarchical modeling strategy to solve the problem of this kind. Attention is limited to the classical first order homogenization in a spatial domain developed here in the framework of a two step (meso–macro) multi-scale computational scheme (FE² problem). Several illustrative examples are presented to investigate the influence of transient flow at the level of constituents (meso-scale) on the macroscopic response including the effect of macro-scale boundary conditions. A two-dimensional section of Charles Bridge subjected to actual climatic conditions is analyzed next to confirm the suitability of algorithmic format of FE² scheme for the parallel computing.     

Mathematics in Context: Measurement,Packaging and Caring for Our Environment

The purpose of this article is to present illustrative examples which will help teachers and teacher educators combine mathematics and science through the activities which involve higher order thinking and worthwhile tasks. Packaging provides an important and meaningful context for exploring important mathematics and an important environmental issue.     

Continuous source in tidal flow: A numerical study of the transport equation

A detailed comparative numerical study of a continuous source discharging in a tidal flow has been performed. The advective term in the one-dimensional transport equation consists of a constant freshwater velocity and an isotropic oscillating component. Various finite element solutions are investigated. Compared to the analytical solution of the dynamic steady state concentration, the numerical results for typical estuarine conditions clearly indicate the superiority of the collocation method with Hermite basis functions. An extended Fourier series analysis that accounts for the source condition is developed to explain the numerical behaviour of the different schemes.     

High order upwind scheme for modelling turbulent shallow water flow in hydraulic structures

An unstructured finite volume model for quasi-2D free surface flow with wet-dry fronts and turbulence modelling is presented. The convective flux is discretised with either a an hybrid second-order/first-order scheme, or a fully second order scheme, both of them upwind Godunov's schemes based on Roe's average. The hybrid scheme uses a second order discretisation for the two unit discharge components, whilst keeping a first order discretisation for the water depth [2]. In such a way the numerical diffusion is much reduced, without a significant reduction on the numerical stability of the scheme, obtaining in such a way accurate and stable results. It is important to keep the numerical diffusion to a minimum level without loss of numerical stability, specially when modelling turbulent flows, because the numerical diffusion may interfere with the real turbulent diffusion. In order to avoid spurious oscillations of the free surface when the bathymetry is irregular, an upwind discretisation of the bed slope source term [4] with second order corrections is used [2]. In this way a fully second order scheme which gives an exact balance between convective flux and bed slope in the hydrostatic case is obtained. The k – ε equations are solved with either an hybrid or a second order scheme. In all the numerical simulations the importance of using a second order upwind spatial discretisation has been checked [1]. A first order scheme may give rather good predictions for the water depth, but it introduces too much numerical diffusion and therefore, it excessively smooths the velocity profiles. This is specially important when comparing different turbulence models, since the numerical diffusion introduced by a first order upwind scheme may be of the same order of magnitude as the turbulent diffusion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)