Geophysical Fluid Dynamics

The workshop “Geophysical Fluid Dynamics” addressed recent advances in analytical, stochastic, modeling and computational studies of geophysical fluid models. Of central interest were the reduced geophysical models, that are derived by means of asymptotic and scaling techniques, and their investigations by methods from the above disciplines. In particular, contributions concerning the viscous and inviscid geostrophic models, the primitive equations of oceanic and atmospheric dynamics, tropical atmospheric models and their coupling to nonlinear dynamics of phase changes moisture, thermodynamical effects, stratifying effects, as well as boundary layers were presented and discussed

Viscous fingering in a radial elastic-walled Hele-Shaw cell

We study the viscous fingering instability in a radial Hele-Shaw cell in which the top boundary has been replaced by a thin elastic sheet. The introduction of wall elasticity delays the onset of the fingering instability to much larger values of the injection flow rate. Furthermore, when the instability develops, the fingers that form on the expanding air-liquid interface are short and stubby, in contrast with the highly-branched patterns observed in rigid-walled cells (Pihler-Puzovi c et al. 2012). We report the outcome of a comprehensive experimental study of this problem and compare the experimental observations to the predictions from a theoretical model that is based on the solution of the Reynolds lubrication equations, coupled to the F oppl-von-K arm an equations which describe the deformation of the elastic sheet. We perform a linear stability analysis to study the evolution of small-amplitude non-axisymmetric perturbations to the time-evolving base flow. We then derive a simpli ed model by exploiting the observations (i) that the non-axisymmetric perturbations to the sheet are very small and (ii) that perturbations to the flow occur predominantly in a small wedge-shaped region ahead of the air-liquid interface. This allows us to identify the various physical mechanisms by which viscous fi ngering is weakened (or even suppressed) by the presence of wall elasticity. We show that the theoretical predictions for the growth rate of small amplitude perturbations are in good agreement with experimental observations for injection flow rates that are slightly larger than the critical flow rate required for the onset of the instability. We also characterize the large-amplitude fingering patterns that develop at larger injection flow rates. We show that the wavenumber of these patterns is still well predicted by the linear stability analysis, and that the length of the fingers is set by the local geometry of the compliant cell.EPSR

Extensions of the Ferry shear wave model for active linear and nonlinear microrheology

The classical oscillatory shear wave model of Ferry et al. [J. Polym. Sci. 2:593-611, (1947)] is extended for active linear and nonlinear microrheology. In the Ferry protocol, oscillation and attenuation lengths of the shear wave measured from strobe photographs determine storage and loss moduli at each frequency of plate oscillation. The microliter volumes typical in biology require modifications of experimental method and theory. Microbead tracking replaces strobe photographs. Reflection from the top boundary yields counterpropagating modes which are modeled here for linear and nonlinear viscoelastic constitutive laws. Furthermore, bulk imposed strain is easily controlled, and we explore the onset of normal stress generation and shear thinning using nonlinear viscoelastic models. For this paper, we present the theory, exact linear and nonlinear solutions where possible, and simulation tools more generally. We then illustrate errors in inverse characterization by application of the Ferry formulas, due to both suppression of wave reflection and nonlinearity, even if there were no experimental error. This shear wave method presents an active and nonlinear analog of the two-point microrheology of Crocker et al. [Phys. Rev. Lett. 85: 888 - 891 (2000)]. Nonlocal (spatially extended) deformations and stresses are propagated through a small volume sample, on wavelengths long relative to bead size. The setup is ideal for exploration of nonlinear threshold behavior

The Sun's Supergranulation

Supergranulation is a fluid-dynamical phenomenon taking place in the solar photosphere, primarily detected in the form of a vigorous cellular flow pattern with a typical horizontal scale of approximately 30--35~megameters, a dynamical evolution time of 24--48~h, a strong 300--400~m/s (rms) horizontal flow component and a much weaker 20--30~m/s vertical component. Supergranulation was discovered more than sixty years ago, however, explaining its physical origin and most important observational characteristics has proven extremely challenging ever since, as a result of the intrinsic multiscale, nonlinear dynamical complexity of the problem concurring with strong observational and computational limitations. Key progress on this problem is now taking place with the advent of 21st-century supercomputing resources and the availability of global observations of the dynamics of the solar surface with high spatial and temporal resolutions. This article provides an exhaustive review of observational, numerical and theoretical research on supergranulation, and discusses the current status of our understanding of its origin and dynamics, most importantly in terms of large-scale nonlinear thermal convection, in the light of a selection of recent findings.Comment: Major update of 2010 Liv. Rev. Sol. Phys. review. Addresses many new theoretical, numerical and observational developments. All sections, including discussion, revised extensively. Also includes previously unpublished results on nonlinear dynamics of convection in large domains, and lagrangian transport at the solar surfac

G-CSC Report 2010

The present report gives a short summary of the research of the Goethe Center for Scientific Computing (G-CSC) of the Goethe University Frankfurt. G-CSC aims at developing and applying methods and tools for modelling and numerical simulation of problems from empirical science and technology. In particular, fast solvers for partial differential equations (i.e. pde) such as robust, parallel, and adaptive multigrid methods and numerical methods for stochastic differential equations are developed. These methods are highly adanvced and allow to solve complex problems.. The G-CSC is organised in departments and interdisciplinary research groups. Departments are localised directly at the G-CSC, while the task of interdisciplinary research groups is to bridge disciplines and to bring scientists form different departments together. Currently, G-CSC consists of the department Simulation and Modelling and the interdisciplinary research group Computational Finance

Mechanisms and Models of Seismic Attenuation

Seismic attenuation is a subject of great interest for both industry and academia. In exploration seismology, wave attenuation must be well understood for interpreting seismic data and laboratory experiments with rocks, and improving the quality and resolution of reflection imaging of the subsurface. To achieve such understanding, mechanisms of seismic attenuation and the associated physical models need to be studied in detail. This dissertation focuses on analyzing several attenuation mechanisms and building first-principle mathematical models for them. The effects of seismic attenuation can be broadly subdivided into two groups: 1) caused by inelasticity of the material and 2) caused by small-scale elastic structures of the material or subsurface. From the first of these groups, I study solid viscosity and internal friction due to squirt flows and wave-induced fluid flows (WIFF) at different scales. This approach is based on a new rheological law called the General Linear Solid (GLS) and recently developed to describe macroscopic inelastic effects in multiphase solids. The GLS is a model composed by time/frequency independent parameters and based on Lagrangian continuum mechanics. By utilizing the GLS framework, I extend the well known-model called the Standard Linear Solid (SLS) to include internal inertial forces, which explains the primary wave and reveals additional highly diffusive wave modes. I also use the GLS to model P-waves with squirt flow dissipation by different configurations of the density, moduli, drag and solid viscosity matrices. Seismic wave attenuation may not only be caused by inelastic properties but also by elastic processes such as reflectivity and scattering. I examine two types of such effects of the elastic structure of the material. First, in a laboratory experiment with several rock types, there is a modest influence of sample size on the measured level of attenuation and modulus dispersion. Second, in a field experiment aimed at measuring Q from seismic reflectivity, the effect of elastic layering can be extremely strong and even completely equivalent to that of the Q. An important general observation from this study is that amplitude decays and phase delays measured from reflection seismic data can always be interpreted as either caused by inelasticity or by small-scale elastic structures. An important complementary goal of studying the mechanisms and effects of seismic attenuation consists in correcting for its effects in seismic records and increasing the resolution of seismic images. In this dissertation, I briefly consider attenuation-correction techniques and develop a novel method for such correction by using time-domain deconvolution. Synthetic and field data are used to illustrate and test the performance of this approach

Viscous fingering instabilities and gravity currents.–—

This thesis examines the possible instability of radially spreading interfaces to the formation of fingers that break the axial symmetry. A well-known example of this occurs when a less viscous fluid displaces a more viscous immiscible fluid either in a porous medium or in a Hele-Shaw cell, which is commonly referred to as the Saffman–Taylor instability. There are three related problems studied in this thesis: a single-layer viscous gravity current spreading from a point source over a rigid surface, radial spreading of an intrusion displacing miscible fluid in a Hele-Shaw cell, and finally, a viscous gravity current spreading from a constant-flux point source over a uniform layer of ambient fluid with equal density but different viscosity. For single-layer viscous gravity currents with constant volumes, an analytical solution is available, which is known to be stable. By means of a numerical linear stability analysis, it is shown here that more general currents, with volumes growing as power laws in time, are stable as well. For currents with constant influx, considering a small shift in temporal origin yields the least stable axisymmetric perturbation mode. This analytic solution is generalised, first to non-axisymmetric perturbations, and then to more general power-law influxes. The derived growth rate confirms theoretically the stability of this least stable mode. Further perturbation modes are found numerically, exploiting a scaling-invariance symmetry of the governing equations, and using a change of independent variable to mitigate the singular nature of the nose. Finally, the stability of a general moving front within the framework of lubrication theory is established by considering the asymptotic limit of large azimuthal wavenumber. Miscible intrusions in a Hele-Shaw cell with negligible diffusion are known to form flat frontal shocks for a sufficiently viscous ambient fluid. Experiments and theoretical work suggest that these fronts become unstable, similar to the Saffman–Taylor instability. However, no formal stability analysis has been done thus far. This thesis caries out this linear stability analysis, showing both that intrusions without a shock are stable and that intrusions with a shock are unstable. An asymptotic analysis of large azimuthal wavenumber shows that the model based on lubrication theory predicts rapidly growing perturbations in this limit. Therefore, the full three-dimensional Stokes equations would be required to predict a most unstable wavenumber. Analytic solutions for the general nonlinear evolution of the intrusion are found in the cases of axisymmetric perturbations and of equal-viscosity fluids. Finally, a viscous gravity current spreading from a constant-flux point source over a uniform layer of ambient fluid is examined for the case of equal-density fluids. This case is identified as a singular limit in which the evolution equation for the interface becomes hyperbolic instead of parabolic. As a consequence, vertical shocks are predicted to form at the front of the intruding current for a sufficiently viscous ambient fluid layer, similar to the shocks found in Hele-Shaw flows. Reintroduction of a small density difference yields an Oleinik entropy condition, which predicts a unique shock height for the self-similar base state. The subsequent linear stability analysis reveals many similarities to Hele-Shaw flows, in particular the singular nature of large azimuthal wavenumbers. Experimental data obtained by others, compares very well overall to predictions of the theory. Finally, the cases of a single-layer current and of a Hele-Shaw intrusion are established as formal asymptotic limits of this two-layer current for large and small influxes, respectively.Engineering and Physical Sciences Research Counci

Vibrational and Structural Characterisation in Two Perovskite Challenges: A Density Functional Theory Study

The modelling of perovskites using density functional theory (DFT) can sometimes be a challenge with many different states very close in energy. In particular, the tilting of the inscribed octahedron, as well as the formation of electron polarons, leads to states with energy differences in the meV range. To distinguish between these states requires special care. This thesis investigates how the vibrational frequencies and defect-induced strain, or chemical expansion, can be used to distinguish between different states. For the polaron state in oxyhydride BaTiO3, the comparison of calculations of hydrogen-ion vibrational frequencies to neutron scattering experiments is an excellent discriminator. The presence of polarons is deemed highly unlikely in unstrained material, despite the presence of oxygen vacancies. The observation is confirmed by comparisons of the strain tensor, calculated using a here-developed formalism. In BaZrO3 the likelihood of an anti-ferrodistortive phase transition is a direct consequence of the magnitude of the R25-mode frequency. The R25-mode frequency is strongly dependent on the lattice spacing, but it is shown that the main effect of the inclusion of gradient corrections, as well as non-local correlation, is secondary and is mostly a consequence of the adjusted lattice constant. The inclusion of Fock exchange, however, leads to a significant stabilisation of the cubic phase, which is also verified by neutron scattering measurements. This thesis also concludes that the inclusion of Fock exchange, as found in hybrid functionals, is essential for a correct description of vibrational properties in both two studied perovskites

Dynamics of the outer planets : 1992 Summer Study Program in Geophysical Fluid Dynamics

The topic this summer was "The Dynamics of the Outer Planets." Andrew Ingersoll gave an excellent review of the current understanding of the strcture of the atmospheres of Jupiter, Neptune, Saturn, and Uranus. He presented the flow structures inferred from the information gathered by the Voyager probes and other observations. The models of the circulations of the interior and of the weather layer - the jets and vortices that we see in the images - were discussed. Jun-Ichi Yano gave further discussions on vortex dynamics in the lab, analytical, and numerical models as applied to the outer planets. Finally, Andy returned with a discussion of thin atmospheres (some so thin that they disappear at night) and new approaches to the dynamics of the interiors. These lectures provided a thorough background in both the data and the theory. As usual, we had talks (or what are sometimes called interactive seminars!) from many visitors during the summer, some directly related to the main topic and others covering other new research in geophysical fluid dynamics. From these, the fellows and staff found new aras for collaborative research and new ideas which they may explore after the summer. Finally, the summer was completed with talks from the fellows on their individual research during the summer. These reports reflect the thought and energy that went into learning new topics and formulating new problems. We look forward to seeing fuller versions of these in journal articles. We gratefully acknowledge the support of the National Science Foundation and the Office of Naval Research. The assistance of Jake Peirson and Barbara Ewing-DeRemer, made the summer, once again, pleasant and easy for all.Funding was provided by the National Science Foundation under Grant No. OCE8901012